If d y d x =x+y x , y(1)=1, then y= - Vidyadip

MCQ

If $\frac{d y}{d x}=\frac{x+y}{x}, y(1)=1$, then $y=$

A
$x+\ln x$
B
$x^2+x \ln x$
C
$x e^{x-1}$
✓
$x+x \ln x$

✓

Answer

Correct option: D.

$x+x \ln x$

It is a homogeneous equation.
Substitute $y=v x$
$\Rightarrow \frac{d y}{d x}=x \frac{d v}{d x}+v$
Now, given equation becomes
$x \frac{d v}{d x}+v=1+v$
$\Rightarrow d v=\frac{d x}{x}$
On integrating both sides, we get
$v=\ln x+c$
$\Rightarrow \frac{y}{x}=\ln x+c$
$\because y(1)=1$
$\Rightarrow x=1, y=1$
$\Rightarrow c=1$
$\therefore y=x+x \ln x$

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