Question

Let $y=f\left(x\right)$ be the solution of the differential equation $(1-x^2)\frac{dy}{dx}-xy=1x\in (-1,1)$.
If $y\left(0\right)=0$ then $y\left(\frac{1}{2}\right)=$

• A. $\frac{\pi }{3\sqrt{3}}$
• B. $\frac{\pi }{\sqrt{3}}$
• C. $\frac{\pi }{6}$
• D. $\frac{\pi }{3}$

Answer 1

The correct option is A $\frac{\pi }{3\sqrt{3}}$

$\frac{dy}{dx}+\frac{(-x)}{1-x^2}y=\frac{1}{1-x^2}I.F.=e^{\int \frac{(-x)}{1-x^2}dx}=\sqrt{1-x^2}y\sqrt{1-x^2}=\int \frac{1}{\sqrt{1-x^2}}dxy\sqrt{1-x^2}={\sin }^{-1}x+c$
$\because y\left(0\right)=0\therefore y\sqrt{1-x^2}={\sin }^{-1}x$