= Solution
The <characteristic polynomial> factors as $(r-2)(r+1)$, so the <complementary solution> is $Ae^{2t}+Be^{-t}$. Each exponential forcing resonates with a homogeneous mode; the <method of undetermined coefficients> therefore uses $te^{2t}$ and $te^{-t}$. Substitution gives the <particular solution> $te^{2t}-te^{-t}-3t$. Thus
$$
y=(A+t)e^{2t}+(B-t)e^{-t}-3t.
$$
The <initial conditions> give $A+B=0$ and $2A-B-3=0$. Hence
$$
\boxed{y(t)=(t+1)e^{2t}-(t+1)e^{-t}-3t}.
$$
The factors of $t$ in the <particular solution> are essential: an unmodified exponential trial lies in the <kernel> of the differential operator.
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