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Explain solution for  RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple choice Question Question 51 maths textbook solution.

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Answer : \text { (d) } \frac{1}{\sqrt{1-y^{2}}}

Hint : The equation is linear in x

Given : \left(1-y^{2}\right) \frac{d y}{d x}+y x=a y,(-1<y<1)

Explanation : Divide on both ides by 1-y^{2}

\Rightarrow \frac{d y}{d x}+\frac{x}{1-y^{2}} y=\frac{a y}{1-y^{2}}

So, the equation becomes linear in x

\begin{aligned} &I F=e^{\int \frac{y}{1-y^{2}} d y} \\ &\text { Let } 1-y^{2}=t \\ &\Rightarrow-2 y d y=d t \\ &\Rightarrow y d y=-\frac{d t}{2} \\ &\Rightarrow I F=e^{-\frac{1}{2} \int \frac{d t}{t}} \end{aligned}

               \begin{aligned} &=e^{-\frac{1}{2} \log t} \\ &=e^{\log t^{-\frac{1}{2}}} \\ &=t^{-\frac{1}{2}} \\ &=\frac{1}{\sqrt{t}} \\ &=\frac{1}{\sqrt{1-y^{2}}} \end{aligned}


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