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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 d2ydx2−2dydx+2y=0y=ex (Acosx+Bsinx)......(1)y1=ex(Acosx+Bsinx)+ex(−Asinx+Bcosx) or y1−y=ex(−Asinx+Bcosx)⋯(2) Differentiating again w.r.t.x, we get y2−y1=ex(−Asinx+Bcosx)+ex(−Acosx−Bsinx) or y2−y1=(y1−y)−y, by (1)and (2)or d2ydx2−2dydx+2y=0, where y1=dydx,y2=d2ydx2

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