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Question

(A) Given the differential equation: $\frac{d^2 y}{d x^2}+y=0$. We check if $y=3 \sin x+4 \cos x$ is a solution. Differentiating with respect to $x$:

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$y=3 \sin x+4 \cos x$

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(A) Given the differential equation: $\frac{d^2 y}{d x^2}+y=0$. We check if $y=3 \sin x+4 \cos x$ is a solution. Differentiating with respect to $x$: $\frac{d y}{d x}=3 \cos x-4 \sin x$. Differentiating again with respect to $x$: $\frac{d^2 y}{d x^2}=-3 \sin x-4 \cos x$. Substituting this into the original equation: $\frac{d^2 y}{d x^2}+y = (-3 \sin x-4 \cos x) + (3 \sin x+4 \cos x) = 0$. Since the equation is satisfied,$y=3 \sin x+4 \cos x$ is a solution. Alternative Method: The characteristic equation for $\frac{d^2 y}{d x^2}+y=0$ is $m^2+1=0$,which gives $m = \pm i$. The general solution is $y=c_1 \cos x+c_2 \sin x$. Option $A$ is of this form with $c_1=4$ and $c_2=3$.

The solution of the differential equation $\frac{d^2 y}{d x^2}+y=0$ is

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