Summary of Nonhomogeneous Linear Equations

Essential Concepts

  • To solve a nonhomogeneous linear second-order differential equation, first find the general solution to the complementary equation, then find a particular solution to the nonhomogeneous equation.
  • Let [latex]y_{p}(x)[/latex] be any particular solution to the nonhomogeneous linear differential equation [latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)[/latex], and let [latex]c_{1}y_{1}(x)+c_{2}y_{2}(x)[/latex] denote the general solution to the complementary equation. Then, the general solution to the nonhomogeneous equation is given by [latex]y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)+y_{p}(x)[/latex].
  • When [latex]r(x)[/latex] is a combination of polynomials, exponential functions, sines, and cosines, use the method of undetermined coefficients to find the particular solution. To use this method, assume a solution in the same form as [latex]r(x)[/latex], multiplying by [latex]x[/latex] as necessary until the assumed solution is linearly independent of the general solution to the complementary equation. Then, substitute the assumed solution into the differential equation to find values for the coefficients.
  • When [latex]r(x)[/latex] is not a combination of polynomials, exponential functions, or sines and cosines, use the method of variation of parameters to find the particular solution. This method involves using Cramer’s rule or another suitable technique to find functions [latex]u^{\prime}(x)[/latex] and [latex]v^{\prime}(x)[/latex] satisfying [latex]\begin{array}{c} \hfill u^{\prime}y_{1}+v^{\prime}y_{2} &= 0 \hfill \\u^{\prime}y_{1}^{\prime}+v^{\prime}y_{2}^{\prime} &= r(x) \hfill\end{array}[/latex]. Then, [latex]y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)[/latex] is a particular solution to the differential equation.

Key Equations

  • Complementary equation
    [latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=0[/latex]
  • General solution to a nonhomogeneous linear differential equation
    [latex]y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)+y_{p}(x)[/latex]

Glossary

complementary equation
for the nonhomogeneous linear differential equation [latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)[/latex] the associated homogeneous equation, called the complementary equation, is [latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=0[/latex]
method of undetermined coefficients
a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess
method of variation of parameters
a method that involves looking for particular solutions in the form [latex]y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)[/latex], where [latex]y_{1}[/latex] and [latex]y_{2}[/latex] are linearly independent solutions to the complementary equations, and then solving a system of equations to find [latex]u(x)[/latex] and [latex]v(x)[/latex].
particular solution
a solution [latex]y_{p}(x)[/latex] of a differential equation that contains no arbitrary constants