13. Implicit differential equations

Lecture

This section is devoted to differential equations not solved with respect to the derivative, which for brevity we
shall call implicit differential equations, by analogy with
implicitly defined functions:
F(x,y,p) = 0, (13.1) p = dy/dx
For simplicity we assume that the function F : R3 → R is smooth and
regular at every point where F = 0 (this last condition can be
dropped, see Remark 13.2). Then in the space J¹ with coordinates (x, y, p) the equation F(x,y,p) = 0 defines a certain smooth surface M, at every point of which a tangent plane is defined.
The main step is the procedure of lifting equation (13.1)
(more precisely, the multivalued field of directions it defines) from the plane (x, y) to the surface M, on which the field of directions becomes
single-valued. This construction is very similar to the use of the Riemann surface for a multivalued function of a complex variable, on which the function becomes single-valued.
13.1. Lifting the equation onto the surface
Let us recall some notions from Section 6.2.6. Let C be an arbitrary curve given in parametric form
x = x(t), y = y(t), (13.2)
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its lift, or 1-graph, is called the curve C'
x = x(t), y = y(t), p = ý(t)/x'(t), (13.3)
in the space J¹. It is easy to see that C is an integral
curve (henceforth we shall for brevity say: a solution) of equation
(13.1) if and only if C lies entirely on the surface M of this
equation (see Fig. 13.1).
Consequently, at every one of its points C is tangent to the tangent plane of the surface M. The equality p = dy/dx means that at every
point of it C' is tangent to the contact plane p dx — dy = 0. Contact planes are defined at every point of the space J¹ and form
in it a field of planes called the contact structure¹.
(x,y) (x,y)
Fig. 13.1 Lifting of solutions of equation (13.1) onto the surface
At almost all points of the surface M the tangent plane does not coincide with the contact plane; at such points their intersection is one-dimensional and therefore defines a field of directions on M. This field is called
the lifted field, its integral curves are 1-graphs of solutions of equation (13.1), and their projections onto the plane (x, y) along the direction of the p-axis are the
solutions of this equation; see Fig. 13.2. Denote the lifted field of directions by A, and the mapping projecting the surface M onto
the plane (x, y) along the direction of the p-axis by π.
The set Σ of points of the surface M at which the projection map π is not a local diffeomorphism (i.e. the set of critical points of the map π) is called the criminant
of the given equation, and its projection π(Σ) the discriminant curve.
The criminant Σ is given by two equations: F = 0, F'p = 0 (the latter
means that at points of the criminant the tangent plane to the surface M is “vertical”²).
Besides the criminant, on the surface M there is one more remarkable curve Σ*, which is called the curve of inflections; it is defined by
the equations F = 0, G = 0, where
G = F'y + pF'x. (13.4)
The meaning of this name and notation will become clear from the following
problems. In Fig. 13.2 the criminant (above) and the discriminant curve
(below) are depicted by long dashed lines, and the curve of inflections Σ* by a dotted line.
(x,y) (x,y)
Fig. 13.2 The lifted field of directions on the surface of equation (13.1)
REMARK 13.1. In the definitions given above the word «curve»
should not be understood literally. In general, the «curve of inflections» and
the «discriminant curve» may be sets that are not curves (the same applies to the criminant). For example, below
we shall encounter an equation whose curve of inflections coincides with
the entire surface of the equation.
However, in the generic case these sets are curves (or
empty), which is why such names have become established. Let us give a sufficient
condition for regularity of the criminant, which will almost always be assumed to hold below.
We shall say that at some point of the criminant of equation (13.1) the regularity condition of the criminant holds, if at this
point³ the rank of the map (x, y, p) → (F, F'p) equals 2, i.e.
rank | F'x F'y F'p | = 2. (13.5)
| F'xp F'yp F'pp |

PROBLEM 13.1. Prove that if at a point T ∈ Σ condition (13.5) holds, then in a neighborhood of T the surface M is regular and the criminant Σ is a regular curve on M. In the case where this condition does not
hold, both the surface and the criminant may have singularities.
PROBLEM 13.2. Prove the following statements:
1. If the function F(x,y,p) is a polynomial in p (with coefficients depending on x, y), then the discriminant curve is given by
the equation D(x,y) = 0, where D is the discriminant of the polynomial F.
2. The tangent plane to the surface M coincides with the contact
plane exactly at those points where the criminant intersects
the curve of inflections. Consequently, the lifted field Δ is defined at
all points of the surface M except the set Σ ∩ Σ*.
3. Points of the curve Σ*, at which F'pp = 0 (i.e. points of the set Σ* \ Σ)
correspond to points of the plane (x, y) at which the solution of the equation
has the form y = f(x), f″ = 0, and, when the additional condition F'ppp ≠ 0 holds, the form y = f(x), f″ = 0, f'″ ≠ 0, i.e. points of a simple
(cubic) inflection of the solution. This explains the name.
HINT FOR PART 3. By the implicit function theorem, in a neighborhood of any point of the set Σ* \ Σ equation (13.1) is solvable
for p, from which it follows that its solution has the form y = f(x).
Then one must differentiate equation (13.1) twice.
Recall that in Section 6.2.7 we applied the Legendre transform to equation (13.1), obtaining as a result the dual equation F*(X,Y,P) = 0, P = dY/dX, and moreover the solutions of these equations
also go over into one another.
PROBLEM 13.3. Prove that under the Legendre transform the criminant of the original equation corresponds to the curve of inflections of the dual equation, and the curve of inflections of the original equation to the criminant of the dual equation. It follows that points of solutions of the original equation lying on the discriminant curve correspond to points of zero curvature of solutions of the dual equation, and, in particular, cusps of solutions of the original equation correspond to points of
simple (cubic) inflection of solutions of the dual equation.
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Let us write an explicit formula for the lifted field of directions Δ. For
this it is convenient to define it by means of a vector field: at every point
where the field Δ is defined, there must be defined a vector generating
this direction (i.e. representing a basis on this line), and
at those points where the direction is not defined, the vector equals zero.
From the algebraic point of view, the intersection of the tangent and contact planes corresponds to the system of equations
F'x dx + F'y dy + F'p dp = 0, (13.6) p dx — dy = 0,
which should be regarded as linear equations in the «differentials» dx, dy, dp — components of a vector from the three-dimensional
tangent space to M at the point (x, y, p).
We shall henceforth always denote derivatives with respect to t by a dot
above: ẋ = dx/dt, ẏ = dy/dt, etc. Then the components of the velocity vector of the curve (13.3), which is the 1-graph of a solution of equation (13.1),
must satisfy the system of equations obtained from (13.6) by replacing the differentials with the corresponding derivatives:
F'x ẋ + F'y ẏ + F'p ṗ = 0, (13.7) p ẋ — ẏ = 0.
From this last system we obtain the lifted vector field
ẋ = F'p, ẏ = p F'p, ṗ = —G, (13.8)
where the function G is defined by formula (13.4).
The lifted field is defined uniquely up to proportionality, i.e. the right-hand sides of equations (13.8) can be multiplied by
any scalar function that does not vanish. The integral
curves of such vector fields coincide geometrically (as sets in phase space); only the speed of motion
along them differs. Thus, to obtain solutions of equation (13.1),
one must project the integral curves of the vector field (13.7)
lying on the surface M onto the plane (x, y).
REMARK 13.2. The construction described carries over without essential change to the case where the surface M is not regular at
certain points forming a nowhere dense set Θ (note that it is contained in the criminant). Although at points of the set Θ
the tangent plane to M is not defined, the lifted field (13.8) extends to them by continuity as zero.
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REMARK 13.3. In practical use of this method one can proceed in two ways. First, one can choose on the surface M two coordinates (out of x, y, p) and write the field (13.8) in
these coordinates, discarding the «redundant» third variable. The resulting one-parameter family of integral curves will be exactly
the set of 1-graphs of solutions of the equation.
For example, if equation (13.1) is equivalent to y = f(x,p), then the variables (x,p) are coordinates on the surface M. The lifted field (13.8)
for the equation f(x,p) — y = 0 on the surface M has the form
ẋ = f'p(x,p), ṗ = f'x(x,p). (13.9)
Integrating the corresponding field of directions dx : dp, we find the family of its integral curves: x = φ(p,c). From this we obtain the solutions
of the original equation (13.1) in parametric form:
x = φ(p,c), y = f(φ(p,c),p).
If, however, it is difficult to choose coordinates on the surface M⁴,
one can consider the field (13.8) in the whole space J¹ and find
the family of all its integral curves. This family depends on
two parameters and contains 1-graphs not only of solutions of equation
(13.1) but of all equations F(x,y,p) = const. To single out solutions of
the equation sought, one must substitute the obtained two-parameter family into (13.1), which allows one to discard extraneous solutions.
EXAMPLE 13.1. For the equation F = p² — 1 = 0 the surface M is a pair of parallel planes p = ±1 in the space
J¹. The lifted vector field (13.8) takes the form
ẋ = 2p, ẏ = 2p², ṗ = 0,
which defines the fields of directions dy : dx = ±1 on the planes p = ±1,
the corresponding families of integral curves are straight lines
y = x + c1 and y = —x + c2. Projecting these families onto the plane
(x, y), we obtain on it an orthogonal net consisting of the lines y = ±x + c (Fig. 13.3). In this example the criminant of the equation
is absent, and the solutions have no singularities, while the curve of inflections contains all points of the surface M (the pair of planes), so that at all
points the solutions have zero curvature. It is obvious that only
straight lines have this last property.
²This also covers the case where the regularity condition of the surface M is violated and it is not the graph of a smooth function.
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p=-1
Fig. 13.3 Illustration for Example 13.1: the integral curves of the lifted
field are projected from the upper and lower planes p = ±1 onto the
coordinate plane (x, y) located between them
13.2. Singularities of projection
At points of the criminant Σ the projection π has a singularity, which is why they are called singular points of equation (13.1). Singular points are divided into two types: regular and irregular.
Regular singular points are defined by the condition G ≠ 0, i.e. these are
points of the set Σ \ Σ*. At them the lifted field of directions is defined
and vertical, as is seen, for example, from formula (13.8). Consequently, through every regular singular point there passes exactly one integral curve of the lifted field — an integral curve of the vector field
(13.8) — and its projection is singular.
Irregular singular points are defined by the condition G = 0, i.e. these are
points of the set Σ ∩ Σ*. At them the tangent and contact planes
coincide, and the lifted field of directions is not defined, while the vector
field (13.8) vanishes. In the generic case irregular singular points occur quite rarely: these are isolated
points of the criminant at which it is crossed by the curve of inflections. We
shall not consider irregular singular points, except for one special case, when they fill the entire criminant
— this leads to the appearance of a singular solution (Section 13.3).
EXAMPLE 13.2. For the equation F = p² — x = 0 the surface
is a parabolic cylinder in the space J¹ with coordinates (y,p) on it. F'p = 2p, so the criminant is the intersection of the cylinder p² — x = 0 with the plane p = 0 (the axis of y on M), and the
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discriminant curve is the y-axis in the plane (x,y). G = —1, so the curve of inflections is absent, and all singular points are regular.
The lifted vector field (13.8) has the form
ẋ = 2p, ẏ = 2p², ṗ = 1.
Integrating the relation dy : dp = 2p², we obtain y = p³/3 + c, which gives
a description of the solutions of the equation in parametric form:
x = p², y = p³/3 + c, c = const. (13.10)
The reader is invited to draw the family of curves (13.10). These are semicubical parabolas, obtained from one of them (for example, at
c = 0) by shifts along the y-axis, lying in the half-plane x ≥ 0. The discriminant curve x = 0 is the locus of their cusps.
The phase portrait arising in Example 13.2 is typical
for a generic equation (13.1). Almost all singular points of such an equation are regular and are fold points of the projection π, which is defined by the condition F'p = 0 and F'pp ≠ 0. Such singular
points are called regular, see Fig. 13.4 (left).
In the generic case the projection map π can
have not only folds of projection but also cusps of projection, and cusp points are regular singular points of equation (13.1).
At such points the criminant Σ has a vertical tangent direction, while the discriminant curve π(Σ) has a cusp at the corresponding
point of the plane. An example of integral curves of the lifted field and their
projections is shown in Fig. 13.4 (right).
REMARK 13.4. In general, cusps of projection can also coincide with irregular singular points of the equation, but this
degeneration has codimension 3 and is therefore unstable.
Indeed, at such points three curves on the surface M intersect: the criminant F'p = 0, the curve of inflections G = 0 and
the curve F'pp = 0. By means of an arbitrarily small perturbation of the function
F the triple intersection of curves can be destroyed, turning it into three
pairwise intersections (see Fig. 9.1 and the discussion in Section 9.1). In this
process one irregular singular point arises on the fold of the projection and one regular singular point at the cusp of the projection, while
the third point of pairwise intersection is not singular.
Below we investigate regular singular points with a fold and with a
cusp (note that in both cases condition (13.5) holds). Let us move the point under consideration to the origin 0 of the space
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J¹, which is done by means of a suitable affine change of coordinates
on the plane (x, y). By definition of a regular singular point, the conditions hold: F(0) = F'p(0) = 0, F'x(0) ≠ 0. By the implicit function
theorem equation (13.1) is locally equivalent to
x = f(y,p) — ... , p = dy/dx, (13.11)
with smooth f, for which f(0) = f'p(0) = 0. Applying Hadamard's lemma,
the function f(y,p) can be written in the form
f(y,p) = h(y) + y p h1(y,p) + p² h2(y,p), h(0) = h'(0) = 0.
Making the change of variable x → x — h(y) in the differential equation (13.11), we obtain a function f(y,p) of the form
f(y,p) = a p² + y p q(y,p) + y² r(y,p), a = ½ f″pp(0). (13.12)
The projection π has the form (p,y) → (x,y), where x = f(y,p). Comparing
this with the map (9.3) of Section 9.1, we see that 0 is a
fold point of projection if a ≠ 0, and a cusp point of projection
if a = 0 and q(0) ≠ 0, r(0) ≠ 0.
13.2.1. Fold of projection
Let us investigate the phase portrait of the equation in a neighborhood of a regular singular point. Without loss of generality one may consider equation (13.11), (13.12) in a neighborhood of the point 0 with a ≠ 0. Solving
(by the implicit function theorem) the equation f'p(y,p) = 0 for
p, we obtain the criminant in the form of the graph of a smooth function p = g(y). Substituting the latter into equation (13.11), we obtain the equation of the discriminant curve in the form of the graph of the function
x = f(y, g(y)). The discriminant curve is smooth, without singularities.
The lifted field of equation (13.11), (13.12) in coordinates (p, y) has the
form ẏ = f'p(y,p), ṗ = 1 — p f'y(y,p), (13.13)
In some neighborhood of the point 0 the second component of the field (13.13) is nonzero, and the vector field (13.13) is transversal to the criminant
p = g(y), so each integral curve of this field intersects
the criminant at a single point.
Consider the integral curve y = φ(p) passing through 0.
This curve is a solution of the equation
dy/dp = f'p(y,p) / (1 — p f'y(y,p)). (13.14)
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Consequently, φ(0) = 0 and φ'(0) = f'p(0) = 0. Further, differentiating
(13.14) with respect to p, we obtain: φ″(0) = f'y(0)φ'(0) + f'pp(0) = 0. Differentiating
once more, we obtain
φ'″'(0) = (f'yy(0)φ'(0) + f'yp(0)) φ″(0) +
+ f'y(0)φ'″'(0) + f'yp(0)φ'(0) + f'ppp(0) = 2a ≠ 0.
From this, taking into account (13.11), (13.12), we obtain
x = f(φ(p),p) = a p² + o(p²), y = φ(p) = ⅓ a p³ + o(p³). (13.15)
From (13.15) it is seen that the solution of equation (13.11), (13.12) passing
through the origin has the form of a semicubical parabola with a cusp at the origin. All other solutions, passing through
nearby points, have a similar form (see Fig. 13.4, left).

fold of projection cusp of projection
Fig. 13.4 Integral curves of the lifted field and their π-projections in the
neighborhood of a fold (left) and a cusp of projection (right)
REMARK 13.5. For regular singular points a stronger statement holds, called the Cibrario theorem⁴, stating that in a neighborhood of a regular singular point all equations
are equivalent to one another, i.e. the families of their solutions are carried into one another
by a suitable local diffeomorphism of the (x, y)-plane.
In particular, in a neighborhood of a regular singular point any equation
(13.1) is equivalent to the equation p² = x considered in Example 13.2.
More on this will be said in Section 13.4 (Theorem 13.2).
⁴In honor of the Italian mathematician Maria Cibrario (Maria Cibrario, 1905–1992).
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13.2.2. Cusp of projection
Now let us investigate the phase portrait of the equation in a neighborhood of a regular singular point which is a cusp of projection.
Without loss of generality one may consider equation (13.11),
(13.12) in a neighborhood of the point 0 with a = 0 and q(0) ≠ 0, r(0) ≠ 0. Applying Hadamard's lemma, let us represent f(y,p) in the form
f(y,p) = a p y + b p³ + p y q1(y,p) + p² r1(y,p), b ≠ 0. (13.16)
From this it is seen that the criminant of the equation f(y,p) — x = 0 has the form
y = k p² + o(p²), where k = —3b/a, and the discriminant curve is
x = f(k p² + o(p²), p) = —2b p³ + o(p³), y = k p² + o(p²),
and it has a cusp at the origin (Fig. 13.4, right). The lifted field in
coordinates (p, y) has the form (13.13) with the function f from (13.16).
PROBLEM 13.4. Show that the integral curves of the lifted
field and their π-projections have the form shown in Fig. 13.5: for k < 0
(left) and for k > 0 (right). These types of singularity are called
respectively the elliptic and hyperbolic cusp. Prove
that at the points of intersection with the discriminant curve (other than the origin
of coordinates), the solutions of the equation have a cusp — a singularity of the form (6.6)
with exponent n = 2, while the solution passing through the origin
has a singularity of the form (6.6) with exponent n = 3.
REMARK 13.6. Looking at the lower part of Fig. 13.5, the attentive reader will recall a similar picture of the section of the
«swallowtail» surface shown in Fig. 11.4 in Section 11.2.4.
In fact, this external similarity points to a deeper
connection. In article [11] it is proved that the family of solutions of any equation in a neighborhood of a regular singular point with a cusp of projection can be obtained from sections of the swallowtail
by a family of parallel planes, shown in Fig. 11.4, by means of
a suitable smooth map R³ → R² of rank 2.
PROBLEM 13.5. Solve the differential equation p³ + a x p = x,
where a is an arbitrary real number, and draw the family of its
solutions depending on the values of a.
HINT. One can use the Legendre transform:
the dual equation is linear and is easily integrated.
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Fig. 13.5 Integral curves of the lifted field and their π-projections:
elliptic cusp (left) and hyperbolic cusp (right)
13.2.3. Singularities of integral curves
In this section we investigate the question of what singularities the solution (integral curve) of equation (13.1), whose 1-graph passes through a regular singular point T0, at which the function F has finite multiplicity in the variable p (see Section 2.1), may have.
For this, condition (13.5) is no longer required. Let
∂¹F/∂p¹(T0) = ... = ∂ṑF/∂pṑ(T0) = 0, ∂ṑ⁺¹F/∂pṑ⁺¹(T0) ≠ 0, (13.17)
and let the desired solution of equation (13.1) have the form (13.2), then its 1-
graph (13.3) is an integral curve of the field (13.8). Without loss of
generality we may assume that this curve passes through T0 at t = 0,
then ẋ(0) = F'p(T0) = 0, ..., i.e. the value t = 0
is critical for both functions x(t), y(t).
Consider the Lie derivative of the vector field (13.8) — the differential operator
L = F'p ∂/∂x + p F'p ∂/∂y — G ∂/∂p
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PROBLEM 13.6. Prove that if ẋ⁽ⁱ⁾(0) = 0 for i = 1, ..., k, then
ẋ⁽Ṍ⁺¹⁾(0) = LṌ(F'p)|₀ = (—p)Ṍ ∂Ṍ⁺¹F/∂pṌ⁺¹|T0 (13.18)
Comparing (13.17) and (13.18), we conclude that x(t) has at zero the
same multiplicity q as F has in p at the point T0:
ẋ(0) = 0, ..., ẋ⁽Ṍ⁾(0) = 0, ẋ⁽Ṍ⁺¹⁾(0) ≠ 0, n = q + 1. (13.19)
Further, as before, we shall for simplicity take the point T0 to be the origin of
coordinates of the space J¹.
PROBLEM 13.7. Prove that if T0 = 0, then the multiplicity of the function
y(t) at zero equals q + 1, i.e.
ẏ(0) = 0, ..., ẏ⁽Ṍ⁾(0) = 0, ẏ⁽Ṍ⁺¹⁾(0) ≠ 0. (13.20)
From formulas (13.19) and (13.20) it follows that the germ of the solution of equation
(13.1) whose 1-graph passes through the regular singular point 0
of finite multiplicity q (formula (13.17)) has the form
x = tⁿ, y = tⁿ⁺¹ + ..., n = q + 1. (13.21)
Germs of curves of this form we studied in detail in Section 6.
PROBLEM 13.8. Is it true that any germ of a curve of the form (13.21)
is a solution of some differential equation?
HINT. One can use the Legendre transform and
pass to the dual equation.
13.3. The phenomenon of singular solutions
The phenomenon of the existence of «singular solutions» was discovered as early as the
18th century. The first known mention of them is contained in a work by Taylor of 1715; the phenomenon was later studied by Euler,
d'Alembert, Clairaut, Cauchy, Laplace, Lagrange, and many others⁵.
In those distant times the primary requirement was to «solve» a differential equation, that is, to describe all its solutions by some
general formula containing an arbitrary constant (the so-called «general solution»). Along this path Taylor once encountered
⁵For the early history of the question see Section 3.4 of the book [28].
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a situation in which, besides the «general solution», there is one more solution, not
obtainable from the general formula for any value of the constant.
For such a solution he introduced the term «singular solution», and this became
the first definition of a singular solution. Taylor's example is contained
in the book [28]; we give a simpler one: the equation p² = y has a family of solutions given by the general formula y = ¼(x + c)², and one
more solution y = 0, not obtainable from it for any c.
Later another definition of a singular solution appeared, based
on a geometric property: a singular solution came to mean a
solution which is the envelope of a family of other (nonsingular)
solutions of the equation⁶. In the equation p² = y given above, the solution
y = 0 is the envelope of the family of the remaining solutions — the parabolas.
For some time it was thought that these two definitions were equivalent
(even Lagrange thought so), but later examples appeared showing that neither of these two conditions follows from the other.
One may note that at every point of the envelope the uniqueness of the solution of the Cauchy problem fails, so that all points of the 1-graph of the envelope are singular points of equation (13.1). Moreover, from what has been said it follows that all these singular points are irregular, i.e.
the 1-graph of the envelope is contained in Σ ∩ Σ* — the intersection of the criminant
and the curve of inflections. This motivates the following definition:
DEFINITION 13.1. A singular solution of equation (13.1) is a solution whose 1-graph consists entirely of irregular singular points. Solutions not satisfying this condition
are called nonsingular.
From what has been said it follows that a singular solution is contained entirely
in the discriminant set of the equation. Let us also note that this
definition includes not only the case of an envelope: for example,
the equation p² = y² has a family of solutions y = c eⁿ, which for
c ≠ 0 are nonsingular, while for c = 0 give the solution y = 0, singular in the sense of
the given definition, but not being an envelope of the family.
REMARK 13.7. Note that in the case of a generic equation (13.1) the set Σ ∩ Σ* is discrete, so a general solution does not
exist. The existence of a singular solution is a rather rare phenomenon,
which occurs only in equations of a special type.
⁶This means that the graph of the singular solution at every point is tangent to some
other (nonsingular) solution, but does not coincide with it identically in any neighborhood of the point.
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Thus, a necessary and sufficient condition for the existence of a singular solution is that the set Σ ∩ Σ* contains a continuous curve whose π-projection is a curve in the
plane (x, y). Let us denote this condition ©.
Two examples of equations of the form (13.1) satisfying condition ©
have already been encountered by us in Section 6.2.7 — these are (6.22) and (6.23). It is also easy
to see that equations of the form p² = f(y) are of this kind: the singular solutions are the lines y = y0, where f(y0) = 0.
The example of the equation p² = y² shows that a singular solution (in the sense of the definition given above) is not always an envelope. Thus,
condition © is necessary but not sufficient for the existence of an envelope of a family of solutions.
THEOREM 13.1. If at a point T ∈ Σ equation (13.1) satisfies condition © and the regularity of the criminant (13.5), then in a neighborhood of this point it has a singular solution which is the envelope of a
family of nonsingular solutions, and at every point the tangency is of
first order. If, moreover, the condition F'yp(T) ≠ 0 holds, then
the singular solution is smooth, otherwise it has a singularity.
PROOF. By means of a suitable affine transformation let us move the point T under consideration to the origin
of the space J¹. Then F'p(0) = 0 and G(0) = F'y(0) = 0 (irregular
singular point), and from condition (13.5) it follows that
F'x(0) ≠ 0, |F'xp(0)| + |F'pp(0)| ≠ 0. (13.22)
By the implicit function theorem the surface M has the form x = f(y,p),
and the criminant is given on it by the equation x = ψ(p) or p = τ(x). We
are in the situation described in Remark 13.3, and the lifted field
for the equation f(x,p) — y = 0⁷ on the surface M has the form (13.9):
ẋ = f'p(x,p), ṗ = p — f'x(x,p).
By virtue of condition © both components of the field (13.9) vanish
on the criminant Σ, which by virtue of condition (13.5) is a curve of the form
x = ψ(p) or p = τ(x). Consequently (see Problem 1.4), we have
ẋ = A(x,p) · u(x,p), ṗ = B(x,p) · u(x,p),
where u = x—ψ(p) or u = p—τ(x) respectively, A and B are certain smooth functions, u(0) = 0. A field with such components can be
divided by the function u(x, p), obtaining the smooth vector field
ẋ = A(x,p), ṗ = B(x,p), A(0) ≠ 0, (13.23)
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whose integral curves coincide with the integral curves
of the field (13.9) everywhere except on the criminant itself. Thus,
all nonsingular solutions of the equation are π-projections of the integral curves of the field (13.23).
PROBLEM 13.9. Prove that at zero the direction of the field (13.23)
equals dp/dx = B/A = —F'xx/F'xp,
where all derivatives are taken at the point 0, and that if it coincides
with the tangent direction of the criminant u(x,p) = 0, this leads to the system
F'xx — ψ' F'xp = 0, ψ' = F'xp/F'pp,
incompatible when conditions (13.22) hold.
Thus, the field (13.23) is transversal to the criminant, and each
of its integral curves transversally intersects Σ at a single
point of the surface M, and its π-projection is a smooth curve having a tangency of first order with the discriminant curve at the corresponding
point of the plane (x, y). The projections of the integral curves of the field (13.23)
form a family of nonsingular solutions of the equation; the discriminant
curve is a singular solution and the envelope of the family of nonsingular
solutions (Fig. 13.6, right).
The last statement, about the regularity of the discriminant curve
when the condition F'yp(T) ≠ 0 holds, is obvious. □
⁷This includes also the case where the regularity condition of the surface M is violated and it is not the graph of a smooth function.
EXAMPLE 13.3. The surface M of the equation F = p² — y = 0 is a parabolic cylinder in the space J¹ with coordinates (x,p). F'p = 2p,
and the criminant is the intersection of the cylinder p² — y = 0 with the plane p = 0
(the x-axis), while the discriminant curve is the x-axis in the plane (x,y).
Since G = —p, the curve of inflections coincides with the criminant, and all
singular points are irregular. The field (13.9) has the form
ẋ = 2p, ṗ = p.
Dividing it by p, we get the field ẋ = 2, ṗ = 1 with the family of integral curves p = x + c. After integrating we obtain
y = ½ p² + c1 p + c2. (13.24)
The two free constants arose because in integrating we did not use the condition p² — y = 0 and found the integral
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typical picture atypical picture
Fig. 13.6 Integral curves of the lifted field and their π-projections for
a regular singular point (left) and an equation with a singular solution —
envelope (right)
curves in the entire space J¹, including those lying on the cylinders p² — y = k with k ≠ 0. Substituting expression (13.24) into the equation
p² — y = 0, we find the relation between the constants: c1 = c2². Substituting this
into (13.24), we obtain the family of nonsingular solutions — parabolas
y = ½ p² + c2² p + c2³ = (½ p + c2)² (13.25)
with envelope y = 0, the discriminant curve of the equation and its singular
solution. The phase portrait is similar to Fig. 13.6 (right).
EXAMPLE 13.4. In Section 6.2.7, using the Legendre transform, we solved the Clairaut equation:
F = f(p) — x p + y = 0, p = dy/dx (13.26)
For it: F'p = f'(p) — x and G = F'y + p F'x = —p + p = 0. The criminant
is given by the equality f'(p) = x, and the «curve of inflections» coincides with the entire
surface M. The lifted field
ẋ = f″(p) — x, ṗ = 0
can be divided by the function f″(p) — x, obtaining the field ẋ = 1, ṗ = 0.
From this it follows that all nonsingular solutions are straight lines. They can be found
by substituting the general equation of a line y = c x + c1 into equation (13.26) and
thereby finding the relation between c and c1.
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But one can proceed differently. As we saw in 6.2.7, the discriminant curve of this equation is dual (in the sense of Legendre) to the graph of
the function Y = f(X) and has the form (6.22). By Theorem 13.1 it is a
singular solution and an envelope of the family of nonsingular solutions. For example, if f(p) = p² (condition F″pp ≠ 0 holds), then the discriminant
curve is a parabola. If f(p) = p³ (condition F″pp ≠ 0 fails
at p = 0), then the discriminant curve is a semicubical parabola,
and the family of its tangent lines is shown in Fig. 6.8.
EXAMPLE 13.5. Consider the equation
F = p² — y p + eⁿ = 0, p = dy/dx (13.27)
for which we have: F'p = 2p — y and G = eⁿ — p². Substituting y = p/2 into
the equality F = 0, we obtain y² = 4eⁿ, and consequently the discriminant set consists of two curves — the graphs of the functions y = ±2eⁿ⁄².
The simplest way is to substitute these functions into equation (13.27) and verify
that they are both (singular) solutions of it. From this it follows, in
particular, that the criminant of the equation consists entirely of irregular singular points, which is also confirmed by direct verification.
The lifted field of equation (13.27) on the surface M has the form
ẋ = 2p — y, ẏ = p(2p—y), ṗ = p² — eⁿ = p² — (y p — p²) = p(2p—y).
Dividing it by 2p — y, we obtain the field
ẋ = 1, ẏ = p, ṗ = p. (13.28)
Integrating dp : dx = p, we obtain p = c eⁿ and y = c eⁿ + c1. This formula gives all the integral curves of the field (13.28) in the space J¹.
In order to exclude the extraneous ones, let us substitute the obtained expressions into equation (13.27). This gives us a relation between the constants: c1 = 1/c. As a result, the family of nonsingular solutions of equation (13.27)
has the form y = c eⁿ + 1/c, c ≠ 0. The singular solutions y = ±2eⁿ⁄² are its envelopes: each curve y = c eⁿ + 1/c is tangent to one of
y = ±2eⁿ⁄² depending on the sign of c. Equation (13.27) has an interesting property, which is discussed below (Remark 14.4).
In conclusion let us consider a few examples of what may
result from a violation of the regularity condition of the criminant.
EXAMPLE 13.6. Below in the table are given three equations satisfying condition ©, but not satisfying the regularity condition of the
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criminant (13.5). In the third column of the table the lifted field is presented and (below it) the same field after cancelling the common factor, and in the
fourth column the family of nonsingular solutions.
The surface M has a singularity: in case 1 it is a pair of planes p = y, and in cases 2 and 3 a semicubical edge of
regression, differently positioned in the space J¹. The criminant is the
x-axis in the space J¹. The discriminant curve y = 0 is the singular solution of the equation in the sense of our definition.
equation lifted field before nonsingular
and after dividing solutions
[table content omitted due to OCR corruption]
In cases 1, 2 the singular solution y = 0 is not the envelope of the family of nonsingular solutions: they do not touch it, but only asymptotically
approach it from both sides (in the first case) or from one
side (in the second). The reason for this is obvious: the field, obtained from the
lifted field by cancelling the common factor, is tangent to the criminant at all its
points, and consequently the criminant is one of its integral curves; therefore it cannot have common points with other
integral curves of this field, i.e. no nonsingular solution can
be tangent to the discriminant curve.
In case 3 the singular solution y = 0 is the envelope of the family of
nonsingular solutions, but at all points the tangency is not of first but of second order. This is a consequence of the fact that each integral curve
of the field obtained from the lifted field by cancelling the common factor intersects the criminant at a single point, but (unlike
Theorem 13.1) not transversally, but with a tangency of first order. The corresponding π-projections have a tangency of order 1 higher.
In the first case the singular solution y = 0 is obtained from the family
of nonsingular solutions at one value of the constant, while in cases 2, 3 it is not.
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In the case of a violation of the regularity condition of the criminant there may be
more complex and interesting phase portraits.
PROBLEM 13.10. Consider the two equations:
p³ + 4xyp² = 8y², (xp² + 3yp — 2y)² = 4y²(yp — 1).
Show that the first equation has a family of nonsingular solutions
y = c(x + c)², which are tangent to the singular solution y = 0 at all points
except the origin, and moreover the singular solution is obtained from this
family at c = 0. Show that the second equation has a family of
nonsingular solutions x = t³ + c t + c1, y = t², tangent to the singular solution
y = 0 at all points except the origin, while the nonsingular solution passing through the
origin has at it a cusp with tangent direction perpendicular to y = 0 (see Example 6.1 and Fig. 6.1 of
Section 6).
13.4. Normal forms
Let us introduce the notion of equivalence of differential equations
of the form (13.1), based on the equivalence of the families of solutions (integral curves) of these equations, namely:
DEFINITION 13.2. Two equations of the form (13.1) will be called
equivalent if there exists a smooth diffeomorphism of the plane (x, y) carrying the family of integral curves of one equation into the family of integral curves of the other. The diffeomorphism by means of which the equivalence is realized is called the conjugating diffeomorphism.
REMARK 13.8. The equivalence thus defined does not reduce to some change of variables turning one equation into another, as is the case
in the class of equations of the form y' = f(x,y). For example, the equations p² = 1 and
p⁴ = 1 have one and the same family of integral curves, and are therefore equivalent, with the conjugating diffeomorphism being the identity map. But it is easy to verify that
neither of these equations can be turned into the other by means of a smooth change of the variables x, y.
In this section we shall establish normal forms (in the sense of the equivalence introduced above) of germs of equations of the form (13.1) at singular
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points of the simplest types. The first type is that of regular singular points,
i.e. those at which F'p = 0, F'pp ≠ 0, F'x ≠ 0.
THEOREM 13.2 (Cibrario). In a neighborhood of a regular singular
point equation (13.1) is equivalent to the equation p² = x, i.e. by means of a suitable local diffeomorphism the family of solutions of
equation (13.1) is brought to the form (13.10).
The proof of this theorem is borrowed (with small changes) from the books [1, 2]. The main change is connected with the fact that we use the division theorem (Theorem 2.1 of Section 2.2), which somewhat
simplifies the proof.
STEP 1. By means of an affine transformation of the plane (x, y)
let us move the point under consideration to the origin of coordinates of the space
J¹. Then F(0) = F'p(0) = 0, F'pp(0) ≠ 0, F'x(0) ≠ 0. From this it follows, in
particular, that the discriminant curve is a regular curve,
and on the plane (x, y) one can choose local coordinates in
which the discriminant curve becomes the axis x = 0. Let us do this
and denote the new coordinates by the same letters. The multiplicity of
the function F(x,y,p) at the point 0 in p equals 1, so for its germ at
the point 0 the representation (2.3) holds with s = 1:
F(x,y,p) = φ(x,y,p) · (p² + 2a(x,y)p + b(x,y)),
where a, b are smooth functions, a(0) = b(0) = 0 and φ(0) ≠ 0. Thus,
in a neighborhood of the point 0 equation (13.1) is equivalent to the equation
p² + 2a(x,y)p + b(x,y) = 0, p = dy/dx. (13.29)
STEP 2. The middle term of equation (13.29) can be killed by a suitable
change y → &tilde{y}, y = u(x,&tilde{y}), where u(0) = 0 and u'ỹ(0) ≠ 0. In this the discriminant curve does not change. Indeed, the change y = u(x,&tilde{y})
turns (13.29) into the equation
(p ũy)² + 2 p ũy (ũx + a(x,u)) + b(x,u) = 0, p = dy/dx, (13.30)
and as u(x, &tilde{y}) one can take the solution of the initial value problem
ũx + a(x,u) = 0, u|ₓ₀ = ψ(&tilde{y}), (13.31)
where ψ(&tilde{y}) is any smooth function such that ψ(0) = 0 and ψ'(0) ≠ 0. The existence of a local solution of problem (13.31) follows from the well-known
theorem for first-order partial differential equations.
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Making such a change and dividing both sides of (13.30) by ũy², we obtain an equation of the form (13.29) with a(x,y) = 0, whose discriminant curve
still coincides with the axis x = 0. From the last it follows that b(0, y) = 0, whence follows the representation b(x, y) = —x c(x, y) with
some smooth c(x,y) (see Problem 1.3, formula (1.6)). Thus, by choosing suitable local coordinates we have brought the original equation (13.1) to the form
p² = x c(x,y), c(0) ≠ 0, (13.32)
whose criminant is the y-axis in the space J¹, and the discriminant curve is the y-axis in the plane (x, y).
STEP 3. Without loss of generality we shall assume that c(0) > 0
(otherwise one must make the change x → —x). Then the surface of equation
(13.32) is a two-sheeted covering of the right half-plane x > 0 with branching along the discriminant curve x = 0, with
the «upper» and «lower» sheets distinguished by the sign:
p = ±√(x c(x,y)), x ≥ 0.
Let us make in the last equation the change x = ξ². One can get rid
of the sign ±, setting √x = ξ, where ξ takes both positive and
negative values. As a result we obtain the equation
dy/dξ = ξ g(ξ², y) (13.33)
with the function g(x, y) = 2√(c(x, y)), whose integral curves intersect the axis ξ = 0 with tangent direction dy : dξ = 0, and at the
points of intersection have a tangency of second order with the lines y = c, i.e. have the form y(ξ) = c + ξ² + o(ξ²). Consequently, the first integral
of equation (13.33) can be represented in the form
I(ξ, y) = y — Φ(ξ, y), Φ(0) ≠ 0.
Let us use Lemma 5.3 of Section 5.3 and represent the function I
in the form I(ξ, y) = φ(ξ², y) + ξ ψ(ξ², y) with smooth φ, ψ. Then,
I(ξ, y) = y — φ(ξ², y) — ξ ψ(ξ², y), ψ(0) ≠ 0. (13.34)
The next step consists in choosing local coordinates in which the first integral (13.34) takes the simplest
form. Before this we offer the reader the following problem.
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PROBLEM 13.11. Explain why the following reasoning is incorrect.
Let us make the change of variable y → Y by the formula Y = I(ξ, y) and turn equation (13.33) into dY/dξ = 0. The integral curves of the latter have the form Y(ξ) = const, which by virtue of x = ξ² gives Y(x) = const,
Thus, equation (13.32) is equivalent to dY/dx = 0.
STEP 4. The change of coordinates (ξ, y) → (η, Y) by the formula
η = ξ ψ(ξ², y), Y = y — φ(ξ², y) (13.35)
is a local diffeomorphism and turns the first integral (13.34) into I(η, Y) = Y — η². In accordance with the change x = ξ², let us introduce in
a similar way the new variable X = η². Then the change (13.35)
corresponds to a change of coordinates (x, y) → (X, Y) given by the formula
X = ξ² ψ²(x,y), Y = y — φ(x, y). (13.36)
The constructed map (x, y) → (X, Y) is a local diffeomorphism. In the new coordinates (X, Y) the integral curves are given by the formula Y — X² = const. By means of a rescaling of the coordinate axes X, Y
this family is brought to the form (13.10), which
means the equivalence of equations (13.32) and p² = x. □
For equations (13.1) satisfying condition ©, there holds
the following analogue of Theorem 13.2:
THEOREM 13.3. If at a singular point T equation (13.1) satisfies condition ©, the regularity of the criminant, and F'yp(T) ≠ 0, then in
a neighborhood of T the equation is equivalent to p² = y, i.e. by means of a suitable local diffeomorphism the family of solutions of equation
(13.1) is brought to the form (13.25).
PROBLEM 13.12. Prove Theorem 13.3 by analogy with Theorem 13.2.
PROBLEM 13.13. From Theorem 13.3 it follows that the Clairaut equation
p² — x p = y in a neighborhood of any singular point is equivalent to the equation p² = y. Prove that here there holds not only local
equivalence but also global equivalence: find (explicitly) a global
diffeomorphism of the plane (x, y), turning the family of solutions
of the first equation into the family of solutions of the second.
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REMARK 13.9. In Theorems 13.2 and 13.3 an important role is played by the fact
that the projection map π has a singularity of «fold» type. In the case of a cusp there are no analogues of these theorems. For example, in article
[16] it is shown that the normal forms of germs of equations (13.1) at regular singular points where the projection has a cusp contain
functional invariants, which makes the classification unsurveyable and thereby vacuous (see Section 8.5).
The reason for the appearance of functional invariants in the case of a cusp
is connected with the fact that on the plane (x, y) there is a region through every
point of which pass three distinct integral curves (see
Figs. 6.8 and 13.5), i.e., as one says, they form a 3-web. In [16] it is shown that the classification of 3-webs has functional invariants
even in the topological category, i.e. if instead of diffeomorphisms
one uses homeomorphisms of the plane.