Question

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

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

Answer 1

Answer: (D)

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