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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 154

Put
$$
a=\frac2{p-1},\qquad s=\log r,
\qquad v(s)=r^au(r).
$$
Then $u=r^{-a}v$ and direct differentiation gives
$$
\Delta u=r^{-a-2}
\left(\ddot v+(d-2-2a)\dot v-a(d-2-a)v\right).
$$
Since $ap=a+2$ and $d-2-2a=2(s_c-1)$, the <Emden-Fowler transformation> turns the equation into
$$
\ddot v+2(s_c-1)\dot v-c_\infty^{p-1}v+v^p=0,
$$
where
$$
c_\infty^{p-1}=a(d-2-a)
=\frac2{p-1}\left(d-2-\frac2{p-1}\right)>0.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 1 / 1 / 2 / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 1 / 1 / 2 / Solution

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 1 / 1 / 2 / Solution

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