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Question

xdydx-y=2y2-x2

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Solution

We have,xdydx-y=2y2-x2dydx=2y2-x2+yxThis is a homogeneous differential equation. Putting y=vx and dydx=v+xdvdx, we getv+xdvdx=2v2x2-x2+vxxv+xdvdx=2v2-1+vxdvdx=2v2-1+v-vxdvdx=2v2-112v2-1dv=1xdxIntegrating both sides, we get12v2-1dv=1xdx1v2-1dv=21xdxlog v+v2-1=2 log x +log Cv+v2-1=Cx2Putting v=yx, we get yx+y2x2-1 =Cx2 y+y2-x2=Cx3Hence, y+y2-x2=Cx3 is the required solution.

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