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

\[\frac{d^2 y}{d x^2} = - A \cos x - B \sin x\]

\[ \Rightarrow \frac{d^2 y}{d x^2} = - \left( A \cos x + B \sin x \right)\]

\[ \Rightarrow \frac{d^2 y}{d x^2} = - y ...........\left[\text{Using equation }\left( 1 \right) \right]\]

⇒ \[\frac{d^2 y}{d x^2} + y = 0\]

Hence, the given function is the solution to the given differential equation.