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 \[\frac{dy}{dx} = 12 \cos3x...........(2)\]

Differentiating both sides of equation (2) with respect to x, we get

\[\frac{d^2 y}{d x^2} = - 36 \sin 3x\]

\[ \Rightarrow \frac{d^2 y}{d x^2} = - 9\left( 4 \sin 3x \right)\]

\[ \Rightarrow \frac{d^2 y}{d x^2} = - 9y ...........\left[\text{ Using equation }\left( 1 \right) \right]\]

⇒ \[\frac{d^2 y}{d x^2} + 9y = 0\]

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