Paper 2, Section II, C

Differential Equations | Part IA, 2009

Consider the first-order ordinary differential equation

dy dx=f1(x)y+f2(x)yp\frac{\mathrm{d} y}{\mathrm{~d} x}=f_{1}(x) y+f_{2}(x) y^{p}

where y0y \geqslant 0 and pp is a positive constant with p1p \neq 1. Let u=y1pu=y^{1-p}. Show that uu satisfies

du dx=(1p)[f1(x)u+f2(x)]\frac{\mathrm{d} u}{\mathrm{~d} x}=(1-p)\left[f_{1}(x) u+f_{2}(x)\right]

Hence, find the general solution of equation ()(*) when f1(x)=1,f2(x)=xf_{1}(x)=1, f_{2}(x)=x.

Now consider the case f1(x)=1,f2(x)=α2f_{1}(x)=1, f_{2}(x)=-\alpha^{2}, where α\alpha is a non-zero constant. For p>1p>1 find the two equilibrium points of equation ()(*), and determine their stability. What happens when 0<p<10<p<1 ?

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