Past exam of the year 1 of the computer science course of the University of Oxford / 2003 / Calculus, probability, linear algebra, discrete mathematics, logic and proof, and models of computation / 1 / a / i / Solution

The repeated root of the characteristic equation suggests factoring out $e^x$. Set $y=e^{x}u$. The product rule givesThus the ordinary differential equation reduces to $u^{\prime \prime }=1$, with general solution $u=x^2/2+Ax+B$. The first initial condition gives $B=1$. Since $y^{\prime }(0)=A+B=2$, the other initial condition gives $A=1$. ThereforeSubstitution into the reduced ordinary differential equation and the two initial conditions verifies the result.