02 - General Solution to a Inhomogeneous Linear ODE

General Solution to a Inhomogeneous Linear ODE

Let $y_{p}x)$ be a solution, aka the particular integral, of the inhomogeneous ODE

$$a_k(x)\frac{d^{k}y}{d{x}^k}+a_{k-1}(x)\frac{d^{k-1}y}{d{x}^{k-1}}+\cdots +a_1(x)\frac{dy}{dx}+a_0(x)y=f(x)$$

Such that $y=y_p$ satisfies the equation above

Then a function $y(x)$ is a solution fo the inhomogeneous linear ODE
If and only if $y(x)$ is in the form of

$$y(x)=y_c(x)+y_p(x)$$

Where $y_c(x)$ is a solution to the corresponding homogenous linear ODE, that is

$$a_k(x)\frac{d^{k}y}{d{x}^k}+a_{k-1}(x)\frac{d^{k-1}y}{d{x}^{k-1}}+\cdots +a_1(x)\frac{dy}{dx}+a_0(x)y=0$$

With $y_c(x)$ being known as the complementary function

Proof

If $y(x)=y_c(x)+y_p(x)$ is a solution to the ODE then

$$a_k(x)\frac{d^k(y_c+y_p)}{d{x}^k}+a_{k-1}(x)\frac{d^{k-1}(y_c+y_p)}{d{x}^{k-1}}+\cdots +a_1(x)\frac{d(y_c+y_p)}{dx}+a_0(x)(y_c+y_p)=f(x)$$

Hence

$$\left(a_k(x)\frac{d^k{y}_c}{d{x}^k}+\cdots +a_0(x)y_c\right)+\left(a_k(x)\frac{d^k{y}_p}{d{x}^k}+\cdots +a_0(x)y_p\right)=f(x)$$

As the second bracket equals $f(x)$ as $y_p(x)$ is a solution to the ODE then
The first bracket must equal $0$ hence

$$y_c(x)\text{ is a solution of the homogeneous ODE}$$

Note that the particular integral is found via guess work (generally similar to $f(x))$