Question

If the solution of the differential equation $x\frac{dy}{dx}+y=x{e}^x$ be $xy=e^x\varphi \left(x\right)+c$ then $\varphi \left(x\right)$ is equal to

• A. $x+1$
• B. $x-1$
• C. $1-x$
• D. $x$

Answer 1

The correct option is B $x-1$

$x\frac{dy}{dx}+y=x{e}^x\frac{dy}{dx}+\frac{y}{x}=e^{x}I.F=e^{\int \frac{1}{x}dx}=xxy=\int x{e}^{x}dxxy=x{e}^x-e^x+c$
comparing with the given solution
$\varphi \left(x\right)=(x-1)$