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  • Need solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.2 question 17

Need solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.2 question 17

Answers (1)

Answer:

The required differential equation is 

xy\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}+x\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}-y\frac{\mathrm{d} y}{\mathrm{d} x}=0

Hint:

 Ellipse centre at the origin and foci on the x-axis

Given:

 Ellipse having the centre at the origin and foci on the x-axis

Solution:

Equation of required ellipse is

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

Where a and b are arbitrary constants

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

\begin{aligned} &\frac{2x}{a^{2}}+\frac{2y}{b^{2}}\frac{\mathrm{d} y}{\mathrm{d} x}=0 \\ &\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(ii) \end{aligned}

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

\begin{aligned} &0=\frac{y}{x}\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}+\frac{\left ( x\frac{\mathrm{d} y}{\mathrm{d} x}-y \right )}{x^{2}}\frac{\mathrm{d} y}{\mathrm{d} x}\\ &xy\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}+x\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}-y\frac{\mathrm{d} y}{\mathrm{d} x}=0 \end{aligned}

The required differential equation is 

xy\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}+x\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}-y\frac{\mathrm{d} y}{\mathrm{d} x}=0

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Gurleen Kaur

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