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  • Provide solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.2 question 16 subquestion (vi)

Provide solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.2 question 16 subquestion (vi)

Answers (1)

Answer:

 The required differential equation is

x\left [ y\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}+\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2} \right ]=y\frac{\mathrm{d} y}{\mathrm{d} x}

Hint:

 This is the equation of hyperbola

\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1

Given:

 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1

Solution:

The equation of family of curves is

\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \qquad \qquad \dots(i)

Where a and b are parameter

Differentiating equation (i) with respect to x, we get

\frac{2x}{a^{2}}-\frac{2y}{b^{2}}\frac{\mathrm{d} y}{\mathrm{d} x}=0 \qquad \qquad \dots(ii)

Differentiating equation (ii) with respect to x, we get

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

Now, from (ii), we get

\begin{aligned} &\frac{2x}{a^{2}}=\frac{2y}{b^{2}}\frac{\mathrm{d} y}{\mathrm{d} x}\\ &\frac{b^{2}}{a^{2}}=\frac{y}{x}\frac{\mathrm{d} y}{\mathrm{d} x} \qquad \qquad \dots (iv) \end{aligned}

From (iii) and (iv), we get

\begin{aligned} &\frac{y}{x}\frac{\mathrm{d} y}{\mathrm{d} x}=\left [ y\left ( \frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}} \right )+\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2} \right ] \end{aligned}

 The required differential equation is

x\left [ y\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}+\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2} \right ]=y\frac{\mathrm{d} y}{\mathrm{d} x}

Posted by

Gurleen Kaur

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