Differential Equations | Part IA, 2004

By writing y(x)=mxy(x)=m x where mm is a constant, solve the differential equation

dydx=x2y2x+y\frac{d y}{d x}=\frac{x-2 y}{2 x+y}

and find the possible values of mm.

Describe the isoclines of this differential equation and sketch the flow vectors. Use these to sketch at least two characteristically different solution curves.

Now, by making the substitution y(x)=xu(x)y(x)=x u(x) or otherwise, find the solution of the differential equation which satisfies y(0)=1y(0)=1.

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