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Question

The differential equation whose general solution is given by, y =
(c1 cos(x+c2))(c3ex+c4)+(c5 sin 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
d3ydx3d2ydx2+dydxy=0
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Solution

The correct option is B d3ydx3+d2ydx2+dydx+y=0
y=c1 cos(x+c2)(c3ex+c4)+(c5sin x)y=c1(cos x cos c2sin x sin c2)(c3 ec4 ex)+(c5 sin x)y=(c1 cos c2)cos x(c1 sin c2c5)sin x(c3 ec4)exy=lcosx+msinxnex............(1)
where l, m, n are arbitrary constant
dydx=l sinx+m cosx+n ex..............(2)d2ydx2=l cosxm sinxn ex.............(3)d3ydx3=l sinxm cosx+n ex..............(4)From equations (1)+(3), d2ydx2+y=2n ex.............(5)From equations (2)+(4), d3ydx3+dydx=2n ex.............(6)From equations (5)+(6), we get d3ydx3+d2ydx2+dydx+y=0

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