Question

The Integrating factor of the differential equation
$(1-y^2)\frac{dx}{dy}+yx=ay$ is

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

Answer 1

The correct option is D $\frac{1}{\sqrt{1-y^2}}$

Given DE is $(1-y^2)\frac{dx}{dy}+yx=ay$

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

which is an exact DE of the form $\frac{dx}{dy}+P\left(y\right)x=Q\left(y\right)$
Integrating factor $IF=e^{\int P\left(y\right)dy}=e^{\int \frac{y}{1-y^2}dy}$
Let $t=1-y^2$

$\Rightarrow dt=-2ydy$
i.e. $IF=e^{\int \frac{-1}{2t}dt}=e^{-\frac{1}{2}\ln t}=\frac{1}{\sqrt{t}}=\frac{1}{\sqrt{1-y^2}}$