Mathematics - Differential Equation Question with Solution | TestHub

Let $y+1=Y$

$\therefore \frac{dY}{dx}=Y^2{e}^{\frac{x^2}{2}}-xY$

Put $-\frac{1}{Y}=k$

$\Rightarrow \frac{dk}{dx}+k\left(-x\right)=e^{\frac{x^2}{2}}$

$I.F.=e^{-\frac{x^2}{2}}$

$\therefore k=\left(x+c\right)e^{x^2/2}$

Put $k=-\frac{1}{y+1}$

$\therefore y+1=-\frac{1}{(x+c)e^{x^2/2}} ...\left(\mathrm{i}\right)$

when $x=2, y=0$, then $c=-2-\frac{1}{e^2}$

Differentiate equation $\left(\mathrm{i}\right) \&$ put $x=1$ we get, 

${\left(\frac{dy}{dx}\right)}_{x=1}=-\frac{e^{3/2}}{{\left(1+e^2\right)}^2}$