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  • Explain solution RD Sharma class 12 chapter Differential Equation exercise 21.2 question 18 maths

Explain solution RD Sharma class 12 chapter Differential Equation exercise 21.2 question 18 maths

Answers (1)

Answer:

The required differential equation is 

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

Hint:

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

Given:

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

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

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

Substituting this value of 1/a2 in equation (ii), we get

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

The required differential equation is 

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

Posted by

Gurleen Kaur

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