Substitute the function y(t) = {\color{red}a} · t + {\color{blue}b} into the ODE y'(t) = C·t + A·y(t) +B.

We obtain y'(t) = \left({\color{red}a} · t + {\color{blue}b}\right)' = {\color{red}a} = C·t + A·({\color{red}a} · t + {\color{blue}b}) +B.

To compare the left- and right-hand side, we rearrange the right-hand side and get

0 · t + {\color{red}a} = (A {\color{red}a} + C) · t + (A {\color{blue}b} +B).

For this to hold for all t, the two equations must be satisfied for the coefficients:

0 = A {\color{red}a}+ C and {\color{red}a} = A {\color{blue}b} +B.

We conclude {\color{red}a} = P and {\color{blue}b} = Q.