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  • Provide solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.2 question 20

Provide solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.2 question 20

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Answer:

 The required differential equation is

\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}=\frac{1}{y}\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}

Hint:

 Differentiating the given equation with respect to x

Given:

 y=ae^{bx+5}

Solution:

y=ae^{bx+5} \qquad \qquad \dots(i)

Where a and b are arbitrary constants

Differentiating with respect to x

\frac{\mathrm{d} y}{\mathrm{d} x}=a.be^{bx+5} \qquad \qquad \dots (ii)

Again, Differentiating with respect to x

\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}=a.b^{2}e^{bx+5} \qquad \qquad \dots (iii)

Put

b=\frac{1}{y}\frac{\mathrm{d} y}{\mathrm{d} x}

from (i) and (ii) in equation (iii), we get

\frac{\mathrm{d}^{2}y }{\mathrm{d} x^{2}}=y.\frac{1}{y^{2}}\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}

\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}=\frac{1}{y}\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}

 The required differential equation is

\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}=\frac{1}{y}\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}

Posted by

Gurleen Kaur

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