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  • Please Solve R.D.Sharma class 12 Chapter 21 Differential Equations Exercise Revision Exercise Question 10 Maths textbook Solution.

Please Solve R.D.Sharma class 12 Chapter 21  Differential Equations Exercise Revision Exercise Question 10 Maths textbook Solution.

Answers (1)

Answer:xy{y}''+x\left ( {y}' \right )^{2}-y{y}'=0

Hint: You must know about the equation of hyperbola

Given: Family of hyperbola having foci on x-axis and centre at origin

Solution: Hyperbola whose foci on y-axis

\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1                       [Two constants, differentiate twice]

\begin{aligned} &\frac{d}{d x}\left[\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right]=\frac{d}{d x}(1) \\ &\frac{1}{a^{2}}\left(\frac{d\left(x^{2}\right)}{d x}\right)-\frac{1}{b^{2}}\left(\frac{d\left(y^{2}\right)}{d x}\right)=0 \\ &\frac{1}{a^{2}}(2 x)-\frac{1}{b^{2}}\left(2 y \frac{d y}{d x}\right)=0 \\ &\frac{2 x}{a^{2}}-\frac{2 y}{b^{2}}\left(\frac{d y}{d x}\right)=0 \end{aligned}

\begin{aligned} &\frac{2 y}{b^{2}}\left(y^{\prime}\right)=\frac{2 x}{a^{2}} \\ &\frac{y}{b^{2}}\left(y^{\prime}\right)=\frac{x}{a^{2}} \\ &\frac{y}{x}\left(y^{\prime}\right)=\frac{b^{2}}{a^{2}} \\ &\frac{y}{x} y^{\prime}=\frac{b^{2}}{a^{2}} \end{aligned}

Again differentiate,

\begin{aligned} &y^{\prime} \frac{\left[\left(y y^{\prime}\right) x-y\right]}{x^{2}}+\frac{y}{x} y^{\prime \prime}=0 . . \text { using product and division rule of differentiation }\\ &\left(y^{\prime} y^{\prime}+y y^{\prime \prime}\right) x-y y^{\prime}=0 . . \text { multiplying by } x^{2}\\ &\left(y^{\prime 2}+y y^{\prime \prime}\right) x-y y^{\prime}=0\\ &x y y^{\prime \prime}+x\left(y^{\prime}\right)^{2}-y y^{\prime}=0 \end{aligned}

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