Question

Show that the function y = A cos x + B sin x is a solution of the differential equation $\frac{d^{2}y}{d{x}^2}+y=0$.

Answer 1

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