I would use partial fraction. What is your final answer?
Use Laplace transforms to solve the initial-value problem,
See figure(s) attached for my work.
I think I started this problem off correctly, however when it comes time to take the inverse laplace transform to obtain y(t) I think I may have taken the wrong route.
Should I be using partial fractions for this or is there another way? (We are a given a table of Laplace transforms that we can readily use without proof)
Let me know what you think.
Thanks again!
Take the Laplace transform gives
This gives
Now comes the hard part, inverting the transform.
First we will use the convolution theorem twice to invert each Fraction. Lets focus on
The inverse tranform of this is
Now we are half way there. Using this we can use the convolution theorem again to invert
This gives
Here is half now you can try the other one.
Maybe I'm missing something here, but don't you get
which implies
Then postulate the partial fraction expansion
add the fractions, set the numerators equal, solve the system of three equations in three unknowns, etc. It's not all that bad, I don't think.
You are on the right track but try this. It will make it easier to find your constants
using what ackbeet said just modify a little
so
now instead of expanding you can force a solution.
lets try s = -2
so
A = 0
Now to find the other constants you can plug in 2 different values for s and make a system of linear equations and solve.
Are Laplace Transforms really necessary here? This is second-order linear constant coefficient non-homogeneous differential equation.
Homogeneous solution:
The characteristic equation is
with multiplicity .
Since the root is repeated, the homogeneous solution is .
Non-homogeneous solution: Assume a solution of the form .
Then
and .
Substituting into your original DE gives
.
Equating like coefficients gives and .
So and .
Therefore, your non-homogeneous solution is
.
So finally, your General solution is the sum of your homogeneous and non-homogeneous solutions...
.