Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 7C / b / Solution

Resolve the forcing in the eigenbasis:In the zero-eigenvalue direction, a constant forcing produces the particular integral $(3a-1)t{u}_1$. In the unit-eigenvalue direction, a constant particular integral is $-(1-2a)u_2$. Thus a particular integral depends on time exactly whenAdding the complementary solution from part (a), the general solution isThis is the eigenvector form of the usual variation of parameters calculation.