 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2005/calculus-linear-algebra-discrete-mathematics-logic-and-proof-and-models-of-computation/3/a/ii/solution

For

y_{1} = e^{x} / x

, on an interval avoiding zero,

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

x y_{1}^{''} + 2 y_{1}^{'} - x y_{1} = e^{x} (1 - \frac{2}{x} + \frac{2}{x ^{2}} + \frac{2}{x} - \frac{2}{x ^{2}} - 1) = 0.

To find an independent solution, put

v = x y

. The product rule gives

v^{''} = x y^{''} + 2 y^{'}

v^{''} - v = 0

. Its independent solutions are

e^{x}

and

e^{- x}

. Hence

y_{2} (x) = \frac{e ^{- x}}{x}, y (x) = \frac{A e ^{x} + B e ^{- x}}{x} (x \neq = 0) .

The two displayed solutions are linearly independent, since their ratio

e^{2 x}

is nonconstant on any open interval. Constants can be chosen separately on the two sides of the singular point

x = 0

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