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Question

Find the differential equations of the family of curves y=ex (Acosx+Bsinx) where A and B are arbitrary constants.

A
d2ydx2+2dydx+2y=0
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B
d2ydx22dydx+2y=0
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C
d2ydx2dydx+2y=0
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D
d2ydx2dydx2y=0
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Solution

The correct option is B d2ydx22dydx+2y=0
y=ex (Acosx+Bsinx)......(1)

y1=ex(Acosx+Bsinx)+ex(Asinx+Bcosx)

or y1y=ex(Asinx+Bcosx)(2)

Differentiating again w.r.t.x, we get
y2y1=ex(Asinx+Bcosx)+ex(AcosxBsinx)

or y2y1=(y1y)y, by (1)and (2)

or d2ydx22dydx+2y=0, where y1=dydx,y2=d2ydx2

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