In each of the following example verify that the given function is a solution of the corresponding differential equation .
$y = c_{1} sin x + c_{2} cos x$ then prove that $\frac{d^{2} y}{d x^{2}} + y = 0$

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Solution

Given,
$y = c_{1} sin x + c_{2} cos x$ .

Now differentiating both sides with respect to $x$ we get,

$\frac{d y}{d x} = c_{1} cos x - c_{2} sin x$

Again differentiating both sides with respect to $x$ we get,

$\frac{d^{2} y}{d x^{2}} = - (c_{1} sin x + c_{2} cos x)$

$\frac{d^{2} y}{d x^{2}} = - y$

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

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In each of the following example verify that the given function is a solution of the corresponding differential equation . y=c1sinx+c2cosx then prove that d2ydx2+y=0

Given,y=c1sinx+c2cosx.Now differentiating both sides with respect to x we get,dydx=c1cosx−c2sinxAgain differentiating both sides with respect to x we get,d2ydx2=−(c1sinx+c2cosx)d2ydx2=−yd2ydx2+y=0.

In each of the following example verify that the given function is a solution of the corresponding differential equation .
$y = c_{1} sin x + c_{2} cos x$ then prove that $\frac{d^{2} y}{d x^{2}} + y = 0$