If $y_1\left(x\right)$  is a solution of the differential equation $\frac{dy}{dx}+f\left(x\right)y=0$ , then a solution of differential equation  $\frac{dy}{dx}+f\left(x\right)y=r\left(x\right)$ is

i) $\frac{d{y}_1}{dx}+f\left(x\right)y_1=0\Rightarrow f\left(x\right)=\frac{-1}{y_1}\frac{d{y}_1}{dx}$

ii) $\frac{dy}{dx}-\frac{1}{y_1}\frac{d{y}_1}{dx}.y=r\left(x\right)$         $e^{-\int \frac{1}{y_1}\frac{dy}{dx}dx}=e^{-\int \frac{d{y}_1}{y_1}}=\frac{1}{y_1}$

$\frac{d}{dx}\left(\frac{y}{y_1}\right)=\frac{r\left(x\right)}{y_1}\Rightarrow \frac{y}{y_1}=\int \frac{r\left(x\right)dx+c}{y_1}$    $y=y_1\int \frac{r\left(x\right)dx}{y_1}+c{y}_1$