Find the convolution between $ e^{-t} u(t) $ and $ e^{-2t} u(t-3)$

Question

Find the convolution between $ e^{-t} u(t) $ and $ e^{-2t} u(t-3)$

I have tried to solve it in the following manner. My answer does not match with the one given in the textbook. I am not sure if I am correct or not.

Solution:

Answer 1

The textbook answer is equivalent to your answer. Observe that $$y(t)=e^{-6}[e^{-(t-3)}u(t)-e^{-2(t-3)}]u(t-3)$$ vanishes if $t\leq 3$. But if $t$ is in this range, then $u(t)=1$. Hence the textbook answer is equivalent to

$$e^{-6}[e^{-(t-3)}-e^{-2(t-3)}]u(t-3)=(e^{-t-3}-e^{-2t})u(t-3),$$ the same as what you obtained.