Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 15 / c

The dependent variable itself does not occur, so reduce order by putting $p=y^{\prime }$. Then $p^{\prime }+2p=3x$, a first-order linear differential equation. Its integrating factor is $e^{2x}$:Integrating once more and renaming the constants givesThe two homogeneous terms are the constant solution and the exponential associated with the roots $0,-2$ of the characteristic equation.

For the general solution in part (i),Thus $B=-3/8$ and $A=11/8$, giving the solution of the initial value problemDirect substitution checks both the differential equation and the two initial conditions.