As mentioned in the problem, the response of
- $ e^{2jt}\, $
is
- $ te^{-2jt}\, $
Suppose we let $ y(t) $ be the response of $ x(t) $, in order to make $ x(t) $ produce the output corresponding to $ y(t) $, we need to multiply the input by $ t $ and make the $ t $ of $ x $ negative. ie.
- $ y(t) = tx(-t)\, $
This can be confirmed by the second condition, which is
- $ te^{2jt}\, $
is the response of
- $ e^{-2jt}\, $
Therefore, using the Euler formula:
- $ cos(2t) = \frac{1}{2}(e^{-2jt}+e^{2jt})\, $
and the two responses mentioned above, the response of $ cos(2t) $ is:
- $ \frac{1}{2}(te^{2jt}+te^{-2jt}) = tcos(2t)\, $
