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Question

Solve the differential equation: ysinxdydx=cosx(sinxy2)

A
y2=23sinx+csin2x
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B
y2=23sinx+csin2x
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C
y2=23sinxcsin2x
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D
y2=23cosx+csin2x
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Solution

The correct options are
A y2=23sinx+csin2x
C y2=23sinxcsin2x
Given differential eqn is
ydydxsinx=cosx(sinxy2)

ydydx+(cotx)y2=cosx
Put y2=z
dzdx=2ydydx
12dzdx+zcotx=cosx
dzdx+2zcotx=2cosx
which is a linear differential eqn with z as dependent variable.
Here, P=2cotx,Q=2cosx
Integrating factor I.F.=ePdx
=e2cotxdx
I.F.=e2logsinx
I.F.=sin2x
Solution of given differential eqn is
zsin2x=2cosxsin2xdx
Put sinx=t
cosxdx=dt
zsin2x=2t33+C
y2sin2x=23sin3x+C
y2=23sinx+Csin2x
C can be positive or negative. Hence, both options are correct

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