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Question

Solution of the differential equation: x+ydydxyxdydx=xsin2(x2+y2)y3

A
cot(x2+y2)=(xy)2+c
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B
y2x2+y2c=tan2(x2+y2)
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C
cot(x2+y2)=(yx)2+c
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D
y2+x2cx2=tan2(x2+y2)
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Solution

The correct options are
A cot(x2+y2)=(xy)2+c
B y2x2+y2c=tan2(x2+y2)

The given differential equation can be written as
xdx+ydyydxxdy=xsin2(x2+y2)y32xdx+2ydyydxxdy=2xsin2(x2+y2)y3d(x2+y2)sin2(x2y2)=2xy3(ydxxdy)cosec2(x2+y2)d(x2+y2)=2(xy)d(xy)
On integrating, we get
cosce2(x2+y2)d(x2+y2)=2(xy)d(xy)
cot(x2+y2)=(xy)2+c
y2x2+y2c=tan2(x2+y2)


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