Differential Equations | Part IA, 2002

Explain what is meant by an integrating factor for an equation of the form

dydx+f(x,y)=0\frac{d y}{d x}+f(x, y)=0

Show that 2yex2 y e^{x} is an integrating factor for

dydx+2x+x2+y22y=0\frac{d y}{d x}+\frac{2 x+x^{2}+y^{2}}{2 y}=0

and find the solution y=y(x)y=y(x) such that y(0)=ay(0)=a, for given a>0a>0.

Show that 2x+x212 x+x^{2} \geqslant-1 for all xx and hence that

dydx1y22y\frac{d y}{d x} \leqslant \frac{1-y^{2}}{2 y}

For a solution with a1a \geqslant 1, show graphically, by considering the sign of dy/dxd y / d x first for x=0x=0 and then for x<0x<0, that dy/dx<0d y / d x<0 for all x0x \leqslant 0.

Sketch the solution for the case a=1a=1, and show that property that dy/dxd y / d x \rightarrow-\infty both as xx \rightarrow-\infty and as xbx \rightarrow b from below, where b0.7035b \approx 0.7035 is the positive number that satisfies b2=ebb^{2}=e^{-b}.

[Do not consider the range xbx \geqslant b.]

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