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