Past exam of the mathematical methods I course of the University of Oxford / 2019 resit / 12 / a

Use variation of parameters, writing $y_p=t_1{u}_1+t_2{u}_2$ and imposing $t_1^{\prime }{u}_1+t_2^{\prime }{u}_2=0$. Differentiation then gives $y_p^{\prime }=t_1{u}_1^{\prime }+t_2{u}_2^{\prime }$. Since the $u_j$ solve the homogeneous equation,Thus solve the two-by-two linear systemIts determinant is the nonzero Wronskian $W$. Cramer's rule gives $t_1^{\prime }=-u_{2}R/W$ and $t_2^{\prime }=u_{1}R/W$, and integration gives the stated antiderivatives. Their integration constants only add homogeneous solutions. The construction applies on a regular interval where the forcing and coefficients are defined and $W\ne 0$.