Question

If $x\,ln\,x \frac{dy}{dx}+y=2\,ln\,x$, $y\left(e\right)=2$, then $y\left(e^{2}\right)=$

• A. $1$
• B. $\frac{3}{2}$
• C. $2$
• D. $\frac{5}{2}$

Answer 1

Answer: (D)

$\frac{dy}{dx}+\frac{y}{x\,ln\,x}=\frac{2}{x}$
It is linear differential equation with
$I.F.=\text{exp}\,\int \frac{dx}{x\,ln\,x}=\,ln\,x$
$\therefore $ Solution is $y\cdot ln\,x=2 \int \frac{ln\,x}{x}dx=\left(ln\,x\right)^{2}+c$
Now, $x=e$,
$y=2$
$\Rightarrow 2=1+c$
$\Rightarrow c=1$
$\Rightarrow y=ln\,x+\frac{1}{ln\,x}$
$\therefore y\left(e\right)^{2}=2+\frac{1}{2}=\frac{5}{2}$.