Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 1 / 1 / 3

Table of contents
- Solution 3

Solution

Seek a homogeneous solution

u (r) = c r^{- β}

. Matching the powers in

Δ u + u^{p} = 0

gives

βp = β + 2

, hence

β = \frac{2}{p - 1} = a .

Since

Δ r^{- β} = - β (d - 2 - β) r^{- β - 2},

matching coefficients gives

c^{p - 1} = β (d - 2 - β) .

Thus the positive Singular homogeneous solution of the Lane-Emden equation is

u_{\infty} (r) = c_{\infty} r^{- 2/ (p - 1)} .

Solved by gpt-5.6-sol high.

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