Homogeneous linear ODE

Definition

$y^{\prime }+p(x)y=r(x)=0$

$\frac{dy}{y}=-p(x)dx$

Solution

General solution is $y={\sum }_{k=1}^n{c}_k{y}_k$ where $\{y_i|i=1,2,\dots ,n\}$ is linearly independent

$\ln |y|=-\int p(x)dx+c$ $\Rightarrow |y|=\exp (-\int p(x)dx)e^C=C\exp (-\int p(x)dx)$ $\Rightarrow y=C\exp (-\int p(x)dx)$

Facts

linear ODE is homogeneous $\iff r(x)=0\iff y=0$ is a solution of linear ODE