#### Provide solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.3 question 2

$y=4 \sin 3 x$  is a solution of  $\frac{d^{2} y}{d x^{2}}+9 y=0$

Hint:

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

Given: $y=4 \sin 3 x$ is a solution

Solution:

Differentiating on both sides,

$\frac{d y}{d x}=4 \cos 3 x(3)$

$\frac{d y}{d x}=12 \cos 3 x$                        ........(i)

Now again differentiating equation (i)

\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}(12 \cos 3 x) \\\\ &\frac{d^{2} y}{d x^{2}}=3(-12 \sin 3 x) \\\\ &\frac{d^{2} y}{d x^{2}}=-36 \sin 3 x \end{aligned}            ........(ii)

Put value of equation (ii) in given problem

\begin{aligned} &\frac{d^{2} y}{d x^{2}}+9 y=0 \\\\ &L H S=-36 \sin 3 x+9(4 \sin 3 x) \end{aligned}

\begin{aligned} &=-36 \sin 3 x+9(4 \sin 3 x) \\\\ &=-36 \sin 3 x+36 \sin 3 x \\\\ &=0 \\\\ &=R H S \end{aligned}

Thus, $y=4 \sin 3 x$ is a solution of $\frac{d^{2} y}{d x^{2}}+9 y=0$.