8C (Differential Equations)

\partial_{x} = τ^{- p} \partial_{η}, \partial_{t} = \partial_{τ} - \frac{p η}{τ} \partial_{η} .

Choose

p = \frac{1}{m + 2}, H (τ) = τ^{- 1/ (m + 2)} .

The PDE then reduces to the stated ODE. One integration, using symmetry at zero, gives

Y^{m} Y^{'} = - η Y

. Hence

Y (η) = [\frac{m}{2} (1 - η^{2})]^{1/ m}, ∣ η ∣ \leq 1,

which has the required endpoint values.

b

For

m = 1

Y = (1 - η^{2}) /2

. Expanding both sides of the perturbation equation gives

(1 - η^{2}) y^{''} - 2 η y^{'} + 6 λ y = 0.

For

y = \sum_{r \geq 0} a_{r} η^{r}

, coefficient comparison gives

a_{r + 2} = \frac{r ( r + 1 ) - 6 λ}{( r + 2 ) ( r + 1 )} a_{r} .

The even and odd series terminate exactly when

6 λ = k (k + 1)

with the corresponding parity. Thus every positive integer

k

gives the regular polynomial solution, a Legendre polynomial up to scale.

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