 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-1/section-b/5/b/solution

On the interval

x > - 1

1 + x

(1 + x)^{- 3}

. Consequently

\frac{d}{d x} [(1 + x)^{- 3} y] = 1 + x, (1 + x)^{- 3} y = \frac{( 1 + x ) ^{2}}{2} + C .

The initial value gives

C = 1

. Thus

y = (1 + x)^{3} [1 + \frac{( 1 + x ) ^{2}}{2}] .

This polynomial also satisfies the original equation at

x = - 1

and gives an extension to all real

x

. The initial value problem is uniquely determined on

x > - 1

; the vanishing leading coefficient at

- 1

does not ensure unique continuation to its left.

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