Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 3 / a

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- Solution a

Solution

Multiply the separable differential equation by

e^{y}

:

e^{y} y^{'} = e^{x}

. The left side is

(e^{y})^{'}

, hence

e^{y} = e^{x} + C

. Therefore

y (x) = ln (e^{x} + C), e^{x} + C > 0.

The real general solution is taken on any connected interval where the logarithm's argument is positive.

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