Differential Equations | Part IA, 2003

(a) Show that if μ(x,y)\mu(x, y) is an integrating factor for an equation of the form

f(x,y)dy+g(x,y)dx=0f(x, y) d y+g(x, y) d x=0

then (μf)/x=(μg)/y\partial(\mu f) / \partial x=\partial(\mu g) / \partial y.

Consider the equation

cotxdytanydx=0\cot x d y-\tan y d x=0

in the domain 0x12π,0y12π0 \leqslant x \leqslant \frac{1}{2} \pi, \quad 0 \leqslant y \leqslant \frac{1}{2} \pi. Using small line segments, sketch the flow directions in that domain. Show that sinxcosy\sin x \cos y is an integrating factor for the equation. Find the general solution of the equation, and sketch the family of solutions that occupies the larger domain 12πx12π,12πy12π-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi,-\frac{1}{2} \pi \leqslant y \leqslant \frac{1}{2} \pi.

(b) The following example illustrates that the concept of integrating factor extends to higher-order equations. Multiply the equation

[yd2ydx2+(dydx)2]cos2x=1\left[y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}\right] \cos ^{2} x=1

by sec2x\sec ^{2} x, and show that the result takes the form ddxh(x,y)=0\frac{d}{d x} h(x, y)=0, for some function h(x,y)h(x, y) to be determined. Find a particular solution y=y(x)y=y(x) such that y(0)=0y(0)=0 with dy/dxd y / d x finite at x=0x=0, and sketch its graph in 0x<12π0 \leqslant x<\frac{1}{2} \pi.

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