Riccati equation of a special type, the general form y '= f (x) y2 + g (x) y + h (x).

Lecture



Special type Riccati equation, general view


y '= f (x) y 2 + g (x) y + h (x). Special type Riccati equation.

1. For a given particular solution, y 0 = y 0 (x), the general solution can be written as

Riccati equation of a special type, the general form y = f (x) y2 + g (x) y + h (x).

where C is some constant. The private solution y 0 = y 0 (x) corresponds to the infinite C.
3. Replacement

Riccati equation of a special type, the general form y = f (x) y2 + g (x) y + h (x).

reduces the general Riccati equation to a second order linear equation:

Riccati equation of a special type, the general form y = f (x) y2 + g (x) y + h (x).

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Terms: HANDBOOK ON MATHEMATICS, SCHOOL MATHEMATICS, HIGHER MATHEMATICS

Math Handbook
Differentiation rules
Derivatives of simple functions, a table of derivatives
Derivatives table, logarithmic and exponential functions
Derivatives table, trigonometric functions
Derivatives table, hyperbolic functions
Integral table, integral of a rational function
The integral table, the integral of the logarithmic function
The integral table, the integral of the exponential function. The improper integral.
Table of integrals, integral of irrational function
Greek alphabet
Latin alphabet
Mathematical constants
Physical constants
Types of algebraic equations
Linear equation definition, examples of solutions
Quadratic equation
Cubic equation
Biquadratic equation
Return (algebraic) equation
Modified and generalized equations of the fourth degree
Generalized return equation
Fourth degree equation of general type
Two-term algebraic equation of nth degree
Special case of equation
Return (algebraic) equation
Algebraic equation of the nth degree of the general form
Types of systems of linear algebraic equations
A system of two linear algebraic equations
A system of m linear algebraic equations
Ordinary differential equations of the first order
Autonomous equation
Equation with separable variables
Linear differential equation
Bernoulli equation
Homogeneous equation
Special type Riccati equation
Special type Riccati equation, case 1
A special type of Riccati equation, case 2 y '= f (x) y2 + ay - ab - b2f (x).
Riccati equation of a special type, case 3 y '= y2 + xf (x) y + f (x).
Special type Riccati equation, case 4 y '= f (x) y2 - axnf (x) y + anxn-1
Special Riccati equation, case 5 y '= f (x) y2 + anxn-1 - a2x2nf (x).
Riccati equation of special type, case 6 y '= - (n + 1) xny2 + xn + 1f (x) y - f (x).
A special type of Riccati equation, case 7 xy '= f (x) y2 + ny + ax2nf (x).
Special Riccati equation, case 8 xy '= x2nf (x) y2 + [axnf (x) - n] y + bf (x).
Riccati equation of a special type, case 9 y '= f (x) y2 + g (x) y - a2f (x) - ag (x).
Special Riccati equation, case 10 y '= f (x) y2 + g (x) y + anxn-1 λ a2x2nf (x) λ axng (x).
A special type of Riccati equation, case 11 y '= aeλxy2 + aeλxf (x) y + λf (x).
Special Riccati equation, case 12 y '= f (x) y2 - aeλxf (x) y + aλeλx.
Riccati equation of special type, case 13 y '= f (x) y2 + aλeλx - a2e2λxf (x).
Special type Riccati equation, case 14 y '= f (x) y2 + λy + ae2λxf (x)
Riccati equation of special type, case 15 y '= y2 - f2 (x) + f' (x).
Special type Riccati equation, case 16 y '= f (x) y2 - f (x) g (x) y + g' (x)
Riccati equation of a special type, the general form y '= f (x) y2 + g (x) y + h (x).
The exponential form of a complex number
Degrees
The roots
Polynomial Horner scheme of the division of P (x) by (x-x0) Bezout theorem
Functions Domain of definition and values ​​Parity and oddness Periodicity Increasing, decreasing function Conversion of graphs of functions
Derivative Rules for calculating derivatives Derivatives table Equation tangent
Newton's formula - Leibniz primitive (indefinite integral). Integral table.
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Real Number Module
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