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Question

Show that y = ex (A cos x + B sin x) is the solution of the differential equation
d2ydx2-2dydx+2y=0.

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Solution

We have,
y=exA cos x+B sin x ...(1)
Differentiating both sides of (1) with respect to x, we get
dydx=exA cos x+B sin x+ex-A sin x+B cos x ...(2)
Differentiating both sides of (2) with respect to x, we get
d2ydx2=exA cos x+B sin x+ex-A sin x+B cos x+ex-A sin x+B cos x+ex-A cos x-B sin xd2ydx2=2ex-A sin x+B cos xd2ydx2=2ex-A sin x+B cos x+2exA cos x+B sin x-2exA cos x+B sin xd2ydx2=2dydx-2y Using 1 and 2d2ydx2-2dydx+2y=0
Hence, the given function is the solution to the given differential equation.

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