Paper 2, Section II, A

(a) The circumference $y$ of an ellipse with semi-axes 1 and $x$ is given by

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

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

(b) Euler proved that $y$ also satisfies the differential equation

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

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

Show that the indicial equation at $t=0$ 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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