 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-2/13/a/solution

r^{2} + r - 2 = (r - 1) (r + 2) = 0

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

A + B = 1

and

A - 2 B = - 1

. Solving yields

A = 1/3

B = 2/3

, hence

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

The two distinct characteristic roots supply all solutions of this second-order linear ordinary differential equation.

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