Mathematics - Differential Equation Question with Solution | TestHub

${\left(x^2+1\right)}^2 \frac{dy}{dx}+2x\left(x^2+1\right)y=1$
$\Rightarrow \frac{dy}{dx}+\frac{2x}{x^2+1}y=\frac{1}{{(x^2+1)}^2}$
Which is a linear differential equation with integrating factor
$$I.F. =$e^{\int \frac{2x}{x^2+1}dx}=e^{\mathrm{l}\mathrm{n}(x^2+1)}=x^2+1$
The solution of the given differential equation will be,
$y.\left(x^2+1\right)=\int \left(x^2+1\right).\frac{1}{{\left(x^2+1\right)}^2}dx=\int \frac{dx}{x^2+1}={\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}x+\mathrm{c}$
Now,$y\left(0\right)= 0$$\Rightarrow \mathrm{C}=0$and$y=\frac{1}{(x^2+1)}{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}x$
$\sqrt{a}y\left(1\right)=\frac{\pi }{32}\Rightarrow \sqrt{a}=\frac{\pi /32}{\pi /8}=\frac{1}{4}$
$\Rightarrow a=\frac{1}{16}$