Question

Particular solution of differential equation $e^{\frac{dy}{dx}}=x;y\left(1\right)=3;x>0$ is ___________.

• A. $logy=x^2+4$
• B. $y=lnx-x+4$
• C. $y^2=logx+4$
• D. $2y=x^2+5$
• E. $y=xlnx-x+4$

Answer 1

The correct option is E $y=xlnx-x+4$

$e^{\frac{dy}{dx}}=x$

Taking log on both sides, we get

$\frac{dy}{dx}=lnx$

$\int dy=\int lnxdx$

$y=xlnx-\int x\frac{1}{x}dx$

$y=xlnx-x+C$

Using the conditon $y\left(1\right)=3$, we get

$3=\left(1\right)ln\left(1\right)-1+C$

$C=4$

So, the particular solution is

$y=xlnx-x+4$

Hence, the answer is option (B).