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Question

21) Show that the following differential equation is homogenous and solve it 2xy+y^2-2x^2.dy/dx

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Solution

dear student

2xy+y2-2x2dydx=0dydx=2xy+y22x2putting y=vxdydx=v+xdvdxdydx=v+xdvdx=2xy+y22x2=2v+v22A first order Differential Equation is Homogeneous when it can be in this form:dydx=fyx=f(v)hence homogeneousv+xdvdx=2v+v22dvv2=dx2x-2v=logx+c-2xy=logx+c

regards

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