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

## General Form

${\displaystyle p(D)y=g(x)}$ , where ${\displaystyle p(D)}$  is a polynomial differential operator of degree ${\displaystyle 2}$  with constant coefficients.

### General Form of the Solution

General solution is of the form ${\displaystyle y(x)=y_{c}(x)+y_{p}(x)}$

where

${\displaystyle y_{c}(x)}$  is called the complimentary solution, and is the solution of associated homogenous equation, ${\displaystyle p(D)y=0}$ .

${\displaystyle y_{p}(x)}$  is called the particular solution, obtained by solving ${\displaystyle p(D)y_{p}=g(x)}$

## Methods to find Complimentary Solution

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

## Methods to find Particular Solution

### Guessing method or method of undetermined coefficients

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

${\displaystyle g(x)}$  ${\displaystyle y_{p}(x)}$
${\displaystyle ae^{bx}}$  ${\displaystyle Ae^{bx}}$
${\displaystyle a\cos {\alpha x}\pm b\sin {\beta x}}$  ${\displaystyle A\cos {\alpha x}\pm B\sin {\beta x}}$
${\displaystyle a_{0}+a_{1}x+\cdots +a_{n}x^{n}}$  ${\displaystyle A_{0}+A_{1}x+\cdots +A_{n}x^{n}}$

Find ${\displaystyle p(D)y_{p}(x)=g(x)}$ , equate coefficients of terms and find the constants ${\displaystyle A}$  and/or ${\displaystyle B}$  and/or ${\displaystyle A_{0},A_{1},\cdots ,A_{n}}$ . If it leads to an undeterminable situation, put ${\displaystyle y_{p}(x)=x\times y_{p}(x)}$  until it’s solvable.

### Variation of parameters

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

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

${\displaystyle y_{p}(x)=-y_{2}(x)\int {y_{1}(x)g(x) \over W_{y_{1},y_{2}}(x)}dx+y_{1}(x)\int {y_{2}(x)g(x) \over W_{y_{1},y_{2}}(x)}dx}$

### Solving by Laplace Transforms

When initial conditions are given,

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

#### Using Convolutions

While solving by Laplace Transforms, if finally ${\displaystyle F(s)}$  is of the form [/itex]g(s)h(s)[/itex], use property of convolutions that

${\displaystyle L{f(t)*g(t)}=L{f(t)}\cdot L{g(t)}}$

and hence ${\displaystyle y(x)=g(s)*h(s)}$ .