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Provide solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.3 question 2

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y=4 \sin 3 x  is a solution of  \frac{d^{2} y}{d x^{2}}+9 y=0


Differentiate the given solution of differential equation on both sides with respect to x

Given: y=4 \sin 3 x is a solution


Differentiating on both sides,

\frac{d y}{d x}=4 \cos 3 x(3)

\frac{d y}{d x}=12 \cos 3 x                        ........(i)

Now again differentiating equation (i)

\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}(12 \cos 3 x) \\\\ &\frac{d^{2} y}{d x^{2}}=3(-12 \sin 3 x) \\\\ &\frac{d^{2} y}{d x^{2}}=-36 \sin 3 x \end{aligned}            ........(ii)

Put value of equation (ii) in given problem

\begin{aligned} &\frac{d^{2} y}{d x^{2}}+9 y=0 \\\\ &L H S=-36 \sin 3 x+9(4 \sin 3 x) \end{aligned}

            \begin{aligned} &=-36 \sin 3 x+9(4 \sin 3 x) \\\\ &=-36 \sin 3 x+36 \sin 3 x \\\\ &=0 \\\\ &=R H S \end{aligned}

Thus, y=4 \sin 3 x is a solution of \frac{d^{2} y}{d x^{2}}+9 y=0.





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