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Question

Solve:
xdx+ydyydxxdy=x2+2y2+y4x2

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Solution

Given, xdx+ydyydxxdy=x2+2y2+y2x2

Now 12d(x2+y2)=xdx+ydy(1)

d(yx)=ydxxdyx2(2)

12d(x2+y2)x2d(yx)=x2+y2+y2+y4x2=x2+y2+y2(1+y2x2)

12d(x2+y2)x2d(yx)=(x2+y2)2x2

d(x2+y2)(x2+y2)2=2d(yx)

Integrating both sides

1x2+y2=2yx+c

2yx+1x2+y2+C=0

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