4.2.2. Inhomogeneous second order linear equations with constant coefficients

Lecture



The general solution of a second-order inhomogeneous linear equation y "+ py '+ qy = f ( x ) is the sum of the general solution y 0 of the homogeneous equation and the particular solution y * of this equation, that is, y = y 0 + y * .

If the right side of the f ( x ) equation is the product of e α · x P n ( x ) , where α is a number, P n ( x ) is a polynomial of degree n , then the particular solution y * is found by selecting the indeterminate coefficients of the polynomial   4.2.2.  Inhomogeneous second order linear equations with constant coefficients degrees n .

  1. If the number α is not the root of the characteristic equation k 2 + pk + q = 0 , then   4.2.2.  Inhomogeneous second order linear equations with constant coefficients .
  2. If α is a single root of the characteristic equation k 2 + pk + q = 0 , then   4.2.2.  Inhomogeneous second order linear equations with constant coefficients
  3. If the number α is the double root of the characteristic equation k 2 + pk + q = 0 , then   4.2.2.  Inhomogeneous second order linear equations with constant coefficients

Let f ( x ) be:

  4.2.2.  Inhomogeneous second order linear equations with constant coefficients
where P ( x ) , R ( x ) are polynomials.

Let n be the greatest degree of these polynomials. The particular solution y * is found by selecting the indefinite coefficients of the polynomials U n ( x ) , V n ( x ) of degree n :
1) if the numbers a ± ib are not the roots of the characteristic equation k 2 + pk + q = 0 , then

  4.2.2.  Inhomogeneous second order linear equations with constant coefficients ;
2) if the numbers a ± ib are the roots of the characteristic equation k 2 + pk + q = 0 , then
  4.2.2.  Inhomogeneous second order linear equations with constant coefficients .

Choosing the form of the particular solution y * corresponding to the right side f ( x ) of the differential equation, we find y * ' , y * "

Substituting the expressions found for y * , y * ' , y * " into the original second-order linear equation, we determine the unknown coefficients of the polynomial   4.2.2.  Inhomogeneous second order linear equations with constant coefficients or the polynomials U ( x ) and V ( x ) .


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Mathematical analysis. Differential equations

Terms: Mathematical analysis. Differential equations