Question

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

• A. $\frac{1}{y\left(x\right)}\int y_1\left(x\right)dx$
• B. $y_1\left(x\right)\int \frac{r\left(x\right)}{y_1\left(x\right)}dx+c$
• C. $intr\left(x\right)y_1\left(x\right)dx$
• D. $noneofthese$

Answer 1

The correct option is B $y_1\left(x\right)\int \frac{r\left(x\right)}{y_1\left(x\right)}dx+c$

$i)\frac{dy}{dx}+f(x)y_1=0\Rightarrow f(x)=\frac{-1}{y_1}\frac{d{y}_1}{dx}ii)\frac{dy}{dx}-\frac{1}{y_1}\frac{d{y}_1}{dx}.y=r\left(x\right)e^{-\int \frac{1}{y_1}\frac{dy}{dx}dx}=e^{-\int \frac{d{y}_1}{dx}}=\frac{1}{y_1}\frac{d}{dx}\left(\frac{y}{y_1}\right)=\frac{r\left(x\right)}{y_1}\Rightarrow \frac{y}{y_1}=\int \frac{r\left(x\right)dx+c}{y_1}y=y_1\int \frac{r\left(x\right)dx}{y_1}+c{y}_1$