Question

Integrating factor of $(1-x^2)\frac{dy}{dx}-xy=1$ is:

• A. $(1-x^2)$
• B. $\sqrt{1-x^2}$
• C. $\frac{1}{1-x^2}$
• D. $\frac{1}{\sqrt{1-x^2}}$

Answer 1

The correct option is B $\sqrt{1-x^2}$

$(1-x^2)\frac{dy}{dx}-xy=1$

$\Rightarrow \frac{dy}{dx}+\frac{-xy}{1-x^2}=\frac{1}{1-x^2}$

Here, $P\left(x\right)=-\frac{x}{1-x^2}$ and $Q\left(x\right)=\frac{1}{1-x^2}$

Then,

$I.F=e^{\int P\left(x\right)dx}=e^{\frac{1}{2}\int \frac{-2x}{1-x^2}dx}=e^{\frac{1}{2}\log (-x^2+1)}=\sqrt{-x^2+1}$

$=\sqrt{1-x^2}$