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#### Provide solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.3 question 6

$y=A e^{B x}$  is a solution of differential equation

Hint:

Differentiate the given solution of differential equation on both sides with respect to $x$

Given:

$y=A e^{B x}$  is a solution.

Solution:

Differentiating on both sides with respect to $x$

$\frac{d y}{d x}=\frac{d}{d x}\left(A e^{B x}\right) \quad\left[\because \frac{d(u v)}{d x}=u \frac{d v}{d x}+v \frac{d u}{d x}\right]$

\begin{aligned} &\frac{d y}{d x}=\left[A \frac{d\left(e^{B x}\right)}{d x}+e^{B x} \frac{d A}{d x}\right] \\\\ &\frac{d y}{d x}=B A e^{B x}+0 \\\\ &\frac{d y}{d x}=B A e^{B x} \end{aligned}    ..............(i)

Now to obtain the second order derivative, differentiate equation (i) with respect to $x$

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

\begin{aligned} &\frac{d^{2} y}{d x^{2}}=B^{2} A e^{B x}+0 \\\\ &\frac{d^{2} y}{d x^{2}}=B^{2} A e^{B x} \end{aligned}        .......(ii)

Now put both equation (i) and (ii) in given differential equation as follows

\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{1}{y}\left(\frac{d y}{d x}\right)^{2} \\\\ &B^{2} A e^{B x}=\frac{1}{A e^{B x}}\left(B A e^{B x}\right)^{2} \end{aligned}

$R H S=\frac{1}{A e^{B x}}\left(B A e^{B x}\right)^{2}$

\begin{aligned} &=\frac{B^{2}\left(A e^{B x}\right)\left(A e^{B x}\right)}{A e^{B x}} \\\\ &=B^{2} A e^{B x} \\\\ &=L H S \end{aligned}

Thus, $y=A e^{B x}$  is a solution of differential equation.