Paper 2, Section II, A

Differential Equations | Part IA, 2011

(a) The circumference yy of an ellipse with semi-axes 1 and xx is given by

y(x)=02πsin2θ+x2cos2θdθ.y(x)=\int_{0}^{2 \pi} \sqrt{\sin ^{2} \theta+x^{2} \cos ^{2} \theta} \mathrm{d} \theta .

Setting t=1x2t=1-x^{2}, find the first three terms in a series expansion of ()(*) around t=0t=0.

(b) Euler proved that yy also satisfies the differential equation

x(1x2)y(1+x2)y+xy=0.x\left(1-x^{2}\right) y^{\prime \prime}-\left(1+x^{2}\right) y^{\prime}+x y=0 .

Use the substitution t=1x2t=1-x^{2} for x0x \geqslant 0 to find a differential equation for u(t)u(t), where u(t)=y(x)u(t)=y(x). Show that this differential equation has regular singular points at t=0t=0 and t=1t=1.

Show that the indicial equation at t=0t=0 has a repeated root, and find the recurrence relation for the coefficients of the corresponding power-series solution. State the form of a second, independent solution.

Verify that the power-series solution is consistent with your answer in (a).

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