STEP III 2007 question 8 solution

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(i)

Putting u = x:

Putting u = exp(-x):

Adding:

General solution:

Applying substitution,

which has general solution

i.e.

y(0) = 0:

(ii)

Applying the substitution gives (detail omitted):

to which "obvious" solutions are u = x, u = exp x, and so the general solution is This gives:

y(0) = 2:

Solution by generalebriety.