Question

The solution of $x\frac{dy}{dx}+y=e^x$ is

• A. $y=\frac{e^x}{x}+\frac{k}{x}$
• B. $y=x{e}^x+cx$
• C. $y=x{e}^x+k$
• D. $x=\frac{e^y}{y}+\frac{k}{y}$

Answer 1

Answer: (A)

$x\frac{dy}{dx}+y=e^x$
$\frac{dy}{dx}+\frac{1}{x}y=\frac{e^x}{x}$
It is a linear differential equation with
$I.F.=e^{\int \frac{1}{x}dx}=e^{logx}=x$
Now, solution is $y\cdot x=\int \frac{e^x}{x}\cdot xdx+k$
$\Rightarrow yx=e^x+k$
$\Rightarrow y=\frac{e^x}{x}+\frac{k}{x}$