 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/discrete-mathematics-continuous-mathematics-and-probability/4/b/i/solution

For

x > 0

, substitute

v = y / x

. Then

y^{'} = v + x v^{'}

, and cancellation reduces the differential equation to

v^{'} + 2 v = e^{- 2 x}

e^{2 x}

gives

(e^{2 x} v)^{'} = 1

. Therefore

y (x) = x (x + C) e^{- 2 x}, C \in R .

Direct substitution gives

y^{'} + 2 y - y / x = x e^{- 2 x}

and verifies the result. Dividing by

x

is legitimate on the stated positive domain.

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