The differential equation whose solution is y | vedclass

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Question

(B) Given the general solution: $y = c_{1} \cos(ax) + c_{2} \sin(ax)$.First,differentiate with respect to $x$: $\frac{dy}{dx} = -a c_{1} \sin(ax) + a

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$\frac{d^{2} y}{d x^{2}} + a^{2} y = 0$

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(B) Given the general solution: $y = c_{1} \cos(ax) + c_{2} \sin(ax)$.First,differentiate with respect to $x$: $\frac{dy}{dx} = -a c_{1} \sin(ax) + a c_{2} \cos(ax)$.Next,differentiate again with respect to $x$: $\frac{d^{2}y}{dx^{2}} = -a^{2} c_{1} \cos(ax) - a^{2} c_{2} \sin(ax)$.Factor out $-a^{2}$: $\frac{d^{2}y}{dx^{2}} = -a^{2} (c_{1} \cos(ax) + c_{2} \sin(ax))$.Since $y = c_{1} \cos(ax) + c_{2} \sin(ax)$,we substitute $y$ into the equation: $\frac{d^{2}y}{dx^{2}} = -a^{2} y$.Rearranging gives: $\frac{d^{2}y}{dx^{2}} + a^{2} y = 0$.

The differential equation whose solution is $y = c_{1} \cos(ax) + c_{2} \sin(ax)$ (where $c_{1}$ and $c_{2}$ are arbitrary constants) is

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