Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 1 / a / Solution

The integrating factor for this linear ordinary differential equation is $\mu (x)=e^{x^3/3}$. Multiplication gives an exact derivative:Consequently $y=1+C{e}^{-x^3/3}$. The initial condition $y(1)=2$ gives $C=e^{1/3}$, so the initial value problem has solutionThe integrating factor calculation determines every solution, and the initial condition fixes its remaining constant of integration uniquely.