Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 13 / b / ii

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- Solution ii

Solution

The real equation has domain

x > 0

. Its integrating factor is

μ (x) = exp (\int (1 + ln x) d x) = e^{x l n x} = x^{x} .

Thus the linear ordinary differential equation becomes

(x^{x} y)^{'} = x^{x} x^{- x} = 1

. After integration,

x^{x} y = x + C

. The initial condition gives

2 = 1 + C

, hence

y (x) = (x + 1) x^{- x}, x > 0.

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