Past exam of the mathematics course of the University of Cambridge / 2009 / ia / Paper 2 / 2C (Differential Equations)

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- Solution 2C

Solution

The characteristic roots of the homogeneous linear ordinary differential equation are

2

and

3

. The forcing is resonant with the

e^{3 x}

solution, so the method of undetermined coefficients uses the particular solution

x e^{3 x}

. Thus

y = A e^{2 x} + B e^{3 x} + x e^{3 x}

. The two initial conditions are

A + B = 0

and

2 A + 3 B + 1 = 0

y = e^{2 x} + (x - 1) e^{3 x} .

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