Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 8 / c

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- Solution c

Solution

Define the Wronskian by

W = y_{1} y_{2}^{'} - y_{1}^{'} y_{2}

. Differentiating and using the normal-form differential equation gives

W^{'} = y_{1} y_{2}^{''} - y_{1}^{''} y_{2} = - P (x) (y_{1} y_{2}^{'} - y_{1}^{'} y_{2}) = - P (x) W .

Integrating this first-order linear differential equation proves Abel's identity:

W (x) = C exp [- \int^{x} p (u) d u], p (u) = \frac{1}{u} + \frac{1}{u - m} .

Thus, on any interval avoiding zero and

m

,

W (x) = \frac{C}{x ( x - m )} .

Absolute-value signs from real logarithms can be absorbed into the constant on each such interval. For linearly independent solutions,

C \neq = 0

.

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