Non-homogeneous second order ODE

#math
Extension to nth order is straightforward
Consider $$y'' + p(x)y' + q(x)y = r(x) \tag{1}$$where $p, q$ are continuous functions. Consider a particular solution $y_{p} (x)$
Let $y$ be a general solution to $(1)$ . Consider $Y = y - y_{p}$ , which satisfies the equation $(1)$ , transforming it into $$Y'' + p(x)Y' + q(x)Y = 0 \tag{2}$$
Clearly we know how to solve for $Y$ , $Y = C_{1} y_{1} + C_{2} y_{2}$ . Thus the general solution to $(1)$ is $$y = C_{1}y_{1} + C_{2}y_{2} + y_{p}$$
We will focus on finding $y_{p}$

Method of variation of parameters

Theorem

A particular solution $y_{p}$ to the linear ODE $(1)$ is given by $$y_{p}(x) = -y_{1}(x)\int\frac{y_{2}(x)r(x)}{W(y_{1},y_{2})}dx + y_{2}(x)\int\frac{y_{1}(x)r(x)}{W(y_{1},y_{2})}dx$$where $W (y_{1}, y_{2}) = y_{1} y_{2}^{'} - y_{1}^{'} y_{2}$ and is referred to as wronskian, $y_{1}, y_{2}$ being basis solutions for the homogeneous counterpart $(2)$

Note

The coefficient of leading term, (highest order differential) must be unity for this method to work.
If it is not, divide the entire equation by it

Non-homogeneous Euler-Cauchy Equation

If the ODE is of the form $$x^2y'' + axy' + by = \bar{r}(x),$$where $a, b$ are constants, then this equation is called non-homogeneous Euler-Cauchy Equation.
We find the solution to the homogeneous counterpart
of this by ignoring $\bar{r} (x)$ , and then we divide it by $x^{2}$ for it to become $$y'' + \frac{a}{x}y' + \frac{b}{x^2}y = r(x)$$and use the Method of variation of parameters to solve for specific solution $y_{p}$ . Thus we can write overall solution as linear combination of $y_{1}, y_{2}, y_{p}$ .