I’ll demonstrate with a couple of examples:
If we use the Laplace transform, we obtain: which leads to . Now we’ve covered how to do this without convolutions. But the convolution integral is much easier: write which means that .
Note: if the integral went too fast for you and you don’t want to use a calculator, use and the integral becomes
Now if we had instead:
The Laplace transform of the equation becomes and hence . One could use the convolution method but partial fractions works easily: one can use the calculator (“algebra” plus “expand”) or:
. Get a common denominator and match numerators:
. One can use several methods to resolve this: here we will use to see which means that and . Now use so obtain which means that so so
So, sometimes the convolution leads us to the answer quicker than other techniques and sometimes other techniques are easier.