Question

The integrating factor of the differential equation $x\frac{dy}{dx}+\left(1+x\right)y=x$ is

• A. $x$
• B. $2x$
• C. $e^x\log x$
• D. $x{e}^x$

Answer 1

The correct option is D

$x{e}^x$

Find the integrating factor for the given differential equation

Given the differential equation, $x\frac{dy}{dx}+\left(1+x\right)y=x$.

$$\begin{gathered} \Rightarrow \frac{dy}{dx}+\frac{1+x}{x}y=1 \\ \Rightarrow \frac{dy}{dx}+\left(\frac{1}{x}+1\right)y=1 \end{gathered}$$

Compare the differential equation with the general form of the linear differential equation $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{+}\mathit{P}\mathit{y}\mathbf{=}\mathit{Q}$.

Here, $P$ and $Q$ are functions of $x$.

Thus, $P=\frac{1}{x}+1$.

So, the integrating factor of the given differential equation can be provided by, $R=e^{\int Pdx}$.

$$\begin{gathered} \Rightarrow R=e^{\int \left(\frac{1}{x}+1\right)dx} \\ \Rightarrow R=e^{\int \left(\frac{dx}{x}+dx\right)}\left[\because \int \frac{1}{x}dx=\log \left(x\right)+c\right] \\ \Rightarrow R=e^{\log x+x} \\ \Rightarrow R=e^{\log x}·e^x \\ \Rightarrow R=x{e}^x \end{gathered}$$

Hence, the integrating factor of the differential equation $x\frac{dy}{dx}+\left(1+x\right)y=x$ is $x{e}^x$, so, the correct answer is option (D).