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  • Explain solution for RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 22 maths textbook solution.

Explain solution for  RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 22 maths textbook solution.

Answers (1)

Answer : \text { (b) } x y y_{2}+x y_{1}^{2}-y y_{1}=0

Hint: Eliminate the constants by differentiating the equation with respect to x

Given: a x^{2}+b y^{2}=1

Explanation: differentiate both sides with respect to x

\begin{aligned} &\Rightarrow 2 a x+2 b y \frac{d y}{d x}=0 \\ &\Rightarrow 2\left(a x+b y \frac{d y}{d x}\right)=0 \\ &\Rightarrow a x+b y \frac{d y}{d x}=0 \end{aligned}                          ....(i)

Again differentiate with respect to x

\begin{aligned} &\Rightarrow a+b\left(\frac{d y}{d x}\right)^{2}+b y \frac{d^{2} y}{d x^{2}}=0 \\ &\Rightarrow b\left(\frac{d y}{d x}\right)^{2}+b y \frac{d^{2} y}{d x^{2}}=-a \\ &\Rightarrow b\left[\left(\frac{d y}{d x}\right)^{2}+y \frac{d^{2} y}{d x^{2}}\right]=-a \end{aligned}

\begin{aligned} &\Rightarrow b\left(y_{1}^{2}+y y_{2}\right)=-a \quad\left[y_{1}=\frac{d y}{d x}, y_{2}=\frac{d^{2} y}{d x^{2}}\right] \\ &\Rightarrow y_{1}^{2}+y y_{2}=-\frac{a}{b} & \ldots(i i) \end{aligned}

Now from (i)

\begin{aligned} &\Rightarrow a x=-b y \frac{d y}{d x} \\ &\Rightarrow y \frac{d y}{d x}=-\frac{a x}{b} \end{aligned}                                    ....(iii)

Compare (ii) and (iii) we get

\begin{aligned} &\Rightarrow y_{1}^{2}+y y_{2}=\frac{y}{x} y_{1} \\ &\Rightarrow x y_{1}^{2}+x y y_{2}=y y_{1} \\ &\Rightarrow x y y_{2}+x y_{1}^{2}-y y_{1}=0 \end{aligned}

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