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Question

The differential equation whose general solution is given by,
y=(c1cos(x+c2))(c3e(x+c4))+(c5sin x) where c1,c2,c3,c4,c5 are arbitrary constants, is

A
d4ydx4d2ydx2+y=0
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B
d3ydx3+d2ydx2+dydx+y=0
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C
d5ydx5+y=0
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D
d3ydx3+d2ydx2+dydxy=0
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Solution

The correct option is B d3ydx3+d2ydx2+dydx+y=0
y=c1cos(x+c2)(c3ex+c4)+(c5sin x)
y=c1(cos xcos c2sin x sin c2)(c3ec4ex)+(c5sin x)
y=(c1cosc2)cos x(c1sinc2c5)sin x(c3ec4)ex
y=1cos x+msin xnex (1)
Where, l, m, n are arbitrary constant
dydx=lsin x+mcos x+nex (2)
d2ydx2=lcosxmsin xnex (3)
d3ydx3=lsin xmcos x+nex (4)
From equations (1) +(3), d2ydx2+y=2nex (5)
From equations (2) +(4), d3ydx3+dydx=2nex (6)
From equations (5) + (6), we get d3ydx3+d2ydx2+dydx+y=0

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