Question

If $x=x(y)$ is the solution of the differential equation $y\frac{dx}{dy}=2x+y^3(y+1)e^y,x(1)=0$; then $x(e)$ is equal to :

• A. $e^3\left(e^e-1\right)$
• B. $e^e\left(e^3-1\right)$
• C. $e^2\left(e^e+1\right)$
• D. $e^e\left(e^2-1\right)$

Answer 1

Answer: (A)

$y\frac{dx}{dy}=2x+y^3(y+1)e^y,x(1)=0$
$\frac{dx}{dy}-\frac{2}{y}x=y^2(y+1)e^y$
I.f $=e^{\int \frac{-2}{y}dy}=\frac{1}{y^2}$
$x\cdot \frac{1}{y^2}=\int (y+1)e^{y}dy$
$\frac{x}{y^2}=(y+1)e^y-e^y+c=y\cdot e^y+c$
$x=y^3{e}^y+c{y}^2$
For $x=0,y=1\Rightarrow c=-e$
$x=y^3{e}^y-e\cdot y^2$
$x(e)=e^3\left(e^e-1\right)$