Use Laplace Transforms to solve the following Initial Value Problems:
(i)
with
(ii)
with , and
There are two ways (at least) of doing this the first involves differentiating one of the equation and then substituting the other into the new equation:
Then solving the new equation for \dot y:
Laplace transforming:
Applying the initial conditions , this becomes:
which you invert by the use of partial fractions and a table of inverse Laplace transforms.
The second method involves writing this as a vector ODE:
where:
and:
Now take the Laplace transform:
Then (as : )
Now with some matrix algebra you can find and and proceed as before.
CB
... use Laplace Transforms to solve the following Initial Value Problems:
with , and
If we intend to apply the LT for resolving a DE, then we have to impose that . So the proposed third order differrential equation must be written as…
... where is the Heavyside unit step function...
Kind regards
No we don't, the default LT in use is the one sided LT and so what the function does at negative arguments is irrelevant to the forward transform.
But the inverse transform is only valid for , since any function that agree on the non-negative half real axis have the same (one-sided) LT.
So the solution obtained using the LT to a IVP is only valid for non-negative times.
CB