4.2.2. Inhomogeneous second order linear equations with constant coefficients

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 degrees n .

1. If the number α is not the root of the characteristic equation k ² + pk + q = 0 , then .
2. If α is a single root of the characteristic equation k ² + pk + q = 0 , then
3. If the number α is the double root of the characteristic equation k ² + pk + q = 0 , then

Let f ( x ) be:

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

;

.

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 or the polynomials U ( x ) and V ( x ) .