Question

Verify that y = 4 sin 3x is a solution of the differential equation $\frac{d^{2}y}{d{x}^2}+9y=0$.

Answer 1

We have,
$y=4\sin 3x$ ...(1)
Differentiating both sides of equation (1) with respect to $x$, we get
$\frac{dy}{dx}=12\cos 3x$ ...(2)
Differentiating both sides of equation (2) with respect to $x$, we get
$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=-36\sin 3x \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=-9\left(4\sin 3x\right) \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=-9y\left[\text{Using equation}\left(1\right)\right] \end{gathered}$$
$\Rightarrow$$\frac{d^{2}y}{d{x}^2}+9y=0$
Hence, the given function is the solution to the given differential equation.