1. Week 9 3. Applications of the LT to ODEs Theorem 1:
If the Laplace transforms of f(t) and f’ (t) exist for some s, then
Proof:
By definition,
Now, integrating by parts with u = e–st and dv = f 'dt (hence, v = f ), obtain...

2. which yields in the desired result.
Theorem 2:
If the Laplace transforms of f(t) and f’’ (t) exist for some s, then
Alternative notation:

3. Algebraic equation for Y(s)
Consider a linear ODE with constant coefficients. It can be solved using the LT as follows:
ODE for y(t)
LT (step 1)
Algebraic equation for Y(s)
solve (step 2)
Y(s) = ...
inverse LT (step 3)
y(t) = ...

4. A quick review of partial fractions:
Consider
where P1 and P2 are polys. in s and the degree of P1 is strictly smaller than that of P2. Assume also that P2 is factorised.
Then...

5. (unrepeated linear factor)
(unrepeated irreducible quadratic factor)
(repeated irreducible quadratic factor)

6. Example 1:
(1)
where
and
(2)
Solution:
Step 0: Observe that
Step 1: Take the LT of (1)...

7. hence, 1 2 1
hence,
Step 2:
Step 3: The inverse LT.

8. Example 2:
Using partial fractions, simplify
Soln: