Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 13 / iii

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- Solution iii

Solution

For the linear differential operator

L = (D - α) (D - β)

, the exponential function is an eigenfunction:

L e^{ϵ x} = (ϵ - α) (ϵ - β) e^{ϵ x}

. Distinct

α, β

give the two homogeneous solutions, while the method of undetermined coefficients supplies a particular solution at the distinct forcing exponent

γ

:

y = C_{1} e^{αx} + C_{2} e^{β x} + \frac{k e ^{γ x}}{( γ - α ) ( γ - β )} .

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