 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2013/12/b/i/solution

On an interval with

x \neq = - 1

(x + 1)^{- 2}

\frac{d}{d x} (\frac{y}{( x + 1 ) ^{2}}) = x + 1.

Integration therefore yields

y = \frac{1}{2} (x + 1)^{4} + C (x + 1)^{2} .

The polynomial formula has a smooth continuation at

x = - 1

, but the original coefficient

2/ (x + 1)

is undefined there. Solution intervals for the stated equation must avoid that point, and the constants on its two sides need not agree.

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