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Information about the mathematical tripos part iii exam held on may 29, 2003, focusing on ordinary differential equations in the complex domain. It includes instructions for attempting three questions out of four, with equal weight, and a warning against starting to read the questions before being instructed by the invigilator. The equations (1) and (2) for a 2x2 system of first order odes, and asks for the determination of pole orders, poincaré ranks, stokes rays, and the number of solutions needed to cover a sector at infinity. It also asks for the formal solution at infinity, the isomonodromic deformation equation, and the compatibility condition between two systems.
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Thursday 29 May 2003 9 to 11
Attempt THREE questions.
There are four questions in total. The questions carry equal weight.
The first three questions relate to the 2 × 2 system of first order ODE’s
dy dλ
= A(λ)y (1)
with A(λ) = A 0 λ^2 + A 1 λ + A 2.
1 What is the order of the pole at ∞?
What is the Poincar´e rank of the singularity λ = ∞?
If A 0 =
determine the Stokes rays.
How many true solutions of given asymptotic behaviour are needed to cover a sector of opening 2π + at ∞? ( > 0, small)
Using the fact that λ = ∞ is the only singularity of the system (1), show that the Stokes matrices S 1 , S 2 ,... , S 6 satisfy the following relation
S 6 S 5 S 4 S 3 S 2 S 1 = 1
Paper 8
