 Solution

ID: past-exam-of-the-differential-equations-and-applications-course-of-the-university-of-oxford/2011/a/1/b/i/solution

The characteristic polynomial after multiplication by four is

r^{2} + 4 r + 4 + π^{2}

, with roots

- 2 \pm iπ

. Thus the general real solution of this constant-coefficient ordinary differential equation is

y (x) = e^{- 2 x} [A cos (π x) + B sin (π x)], A, B \in R .

y = e^{- 2 x} v

gives

L y = \frac{1}{4} e^{- 2 x} (v^{''} + π^{2} v)

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