# Ordinary Differential Equations:Cheat Sheet/Second Order Inhomogeneous Ordinary Differential Equations

## General Form[edit | edit source]

, where is a polynomial differential operator of degree with constant coefficients.

### General Form of the Solution[edit | edit source]

General solution is of the form

where

is called the complimentary solution, and is the solution of associated homogenous equation, .

is called the particular solution, obtained by solving

## Methods to find Complimentary Solution[edit | edit source]

Methods to solve for complimentary solution is discussed in detail in the article Second Order Homogeneous Ordinary Differential Equations.

## Methods to find Particular Solution[edit | edit source]

### Guessing method or method of undetermined coefficients[edit | edit source]

Choose appropriate y_p (x) with respect to g(x) from table below:

Find , equate coefficients of terms and find the constants and/or and/or . If it leads to an undeterminable situation, put until it’s solvable.

### Variation of parameters[edit | edit source]

This method is applicable for inhomogeneous ODE with variable coefficients in one variable.

Suppose two linearly independent solutions of the ODE are known. Then

### Solving by Laplace Transforms[edit | edit source]

When initial conditions are given,

- Find Laplace Transform of either sides (See notes in earlier chapter for few common transforms)
- Isolate F(s)
- Split R.H.S. into partial fractions
- Find inverse Laplace Transforms.

#### Using Convolutions[edit | edit source]

While solving by Laplace Transforms, if finally is of the form </math>g(s)h(s)</math>, use property of convolutions that

and hence .