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# The integrating factor of the differential equation $(1-y^2)\large\frac{dx}{dy}$$+yx=ay\: \;is $(A)\;\frac{1}{y^2-1}\qquad(B)\;\frac{1}{{\sqrt{y^2-1}}}\qquad(C)\;\frac{1}{1-y^2}\qquad(D)\;\frac{1}{{\sqrt{1-y^2}}}$ Can you answer this question? ## 1 Answer 0 votes Toolbox: • The integrating factor of the equation \large\frac{dx}{dy}$$ + px = Q$ is $e^{\large\int p dy}$
Step 1:
Let us rewrite the equation as $\large\frac{dx}{dy} +\frac{ yx}{(1+y^2)} =\frac{ ay}{(1-y^2)}$
Here $p=\large\frac{y}{(1+y^2)}$
Step 2:
$\int \large\frac{y}{(1-y^2)}$$dy=\large\frac{1}{2}$$\log(1-y^2)=-\log\sqrt{1-y^2}$
Hence the integration factor is $e^{\log\sqrt{\large\frac{1}{\mid1-y^2\mid}}}$
$\Rightarrow \large\frac{1}{\sqrt{1-y^2}}$
Hence the correct option is D.