Past exam of the mathematical methods I course of the University of Oxford / 2013 / 12 / a / Solution

The second equation gives $y=t+x^{\prime }-x$. Differentiating it gives $y^{\prime }=1+x^{\prime \prime }-x^{\prime }$. Substitution into the first equation cancels the $x^{\prime }$ terms and the constant forcing, leaving $x^{\prime \prime }+x=0$. HenceThis has two independent constants, as expected for the first-order linear system of differential equations. At zero, the conditions give $A=0$ and $B=1$. Thus the required particular solution isChecking directly, $x^{\prime }+y^{\prime }+x=\cos t+(1-\sin t-\cos t)+\sin t=1$, while $-x^{\prime }+x+y=t$. This also verifies the signs introduced during elimination.