The solution of x d y d x +y=e^x is - Vidyadip

MCQ

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

✓
$y=\frac{e^x}{x}+\frac{c}{x}$
B
$y=x e^x+c x$
C
$y=x e^x+c$
D
$x=\frac{e^y}{y}+\frac{c}{y}$

✓

Answer

Correct option: A.

$y=\frac{e^x}{x}+\frac{c}{x}$

$x \frac{d y}{d x}+y=e^x$
$\frac{d y}{d x}+\frac{y}{x}=\frac{e^x}{x}$
It is a linear differential equation with
$\text { I.F. }=e^{\int \frac{1}{x} d x}=e^{\log x}=x$
Now, solution is $y \cdot x=\int \frac{e^x}{x} \cdot x d x+c$
$\Rightarrow y x=e^x+c$
$\Rightarrow y=\frac{e^x}{x}+\frac{c}{x}$

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