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

General Form
$$p(D)y=g(x)$$, where $$p(D)$$ is a polynomial differential operator of degree $$2$$ with constant coefficients.

General Form of the Solution
General solution is of the form $$y(x)=y_c (x)+y_p (x)$$

where

&emsp;$$y_c (x)$$ is called the complimentary solution, and is the solution of associated homogenous equation, $$p(D)y=0$$.

&emsp;$$y_p (x)$$ is called the particular solution, obtained by solving $$p(D) y_p=g(x)$$

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

Guessing method or method of undetermined coefficients
Choose appropriate y_p (x) with respect to g(x) from table below:

Find $$p(D) y_p (x)=g(x)$$, equate coefficients of terms and find the constants $$A$$ and/or $$B$$ and/or $$A_0,A_1,\cdots,A_n$$. If it leads to an undeterminable situation, put $$y_p (x)=x\times y_p (x)$$ 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

&emsp;$$y_p (x)=-y_2 (x)\int{y_1 (x)g(x)\over W_{y_1,y_2} (x) }dx+y_1 (x)\int{y_2 (x)g(x)\over W_{y_1,y_2} (x) }dx$$

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 $$F(s)$$ is of the form g(s)h(s), use property of convolutions that

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

and hence $$y(x)=g(s)*h(s)$$.