Mathematics - Differential Equation Question with Solution | TestHub

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

(ii) $\frac{dy}{dx}-\frac{1}{y_1}\frac{d{y}_1}{dx}\cdot y=r(x)$

$e^{-\int \frac{1}{y_1}\frac{dy}{dx}dx}=e^{-\int \frac{d{y}_1}{y_1}}=\frac{1}{y_1}$

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