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

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

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