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  • Please solve RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 30 maths textbook solution.

Please solve RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 30 maths textbook solution.

Answers (1)

Answer : (a)\left(x^{2}-1\right)=C\left(y^{2}+1\right)

Hint: Use variable separable method i.e. separate x and y terms by taking x terms in one side and y terms to the other.

Given : x d x+y d x=x^{2} y d y-y^{2} x d x

Explanation  : x d x+y^{2} x d x=x^{2} y d y-y d y

\begin{aligned} &\Rightarrow x\left(1+y^{2}\right) d x=y\left(x^{2}-1\right) d y \\ &\Rightarrow \frac{x}{x^{2}-1} d x=\frac{y}{y^{2}+1} d y \end{aligned}

Integrate both sides

\begin{aligned} &\Rightarrow \int \frac{x}{x^{2}-1} d x=\int \frac{y}{y^{2}+1} d y\\ &\text { Let } x^{2}-1=t \text { and } y^{2}+1=v \end{aligned}

\begin{aligned} &\Rightarrow 2 x d x=d t \text { and } 2 y d y=d v \\ &\Rightarrow x d x=\frac{d t}{2} \text { and } y d y=\frac{d v}{2} \end{aligned}

Put in (i)

\begin{aligned} &\Rightarrow \frac{1}{2} \int \frac{d t}{t}=\frac{1}{2} \int \frac{d v}{v} \\ &\Rightarrow \frac{1}{2} \log t=\frac{1}{2} \log v+\frac{1}{2} \log C \end{aligned}

\begin{aligned} &\Rightarrow \log t=\log v+\log C \\ &\Rightarrow \log t=\log C v \end{aligned}

\begin{aligned} &\Rightarrow t=C v \\ &\Rightarrow\left(x^{2}-1\right)=C\left(y^{2}+1\right) \end{aligned}

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