How do you find the solution using the Laplace transform?
The solution is accomplished in four steps:
- Take the Laplace Transform of the differential equation. We use the derivative property as necessary (and in this case we also need the time delay property)
- Put initial conditions into the resulting equation.
- Solve for Y(s)
- Get result from the Laplace Transform tables. (
How do Laplace transforms work?
The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform.
How do you solve an IVP with Laplace transforms?
There are a couple of things to note here about using Laplace transforms to solve an IVP. First, using Laplace transforms reduces a differential equation down to an algebra problem. In the case of the last example the algebra was probably more complicated than the straight forward approach from the last chapter.
Why is the Laplace transform used to solve differential equations?
When such a differential equation is transformed into Laplace space, the result is an algebraic equation, which is much easier to solve. Furthermore, unlike the method of undetermined coefficients, the Laplace transform can be used to directly solve for functions given initial conditions.
Why do we need initial values for the Laplace transform?
This is because we need the initial values to be at this point in order to take the Laplace transform of the derivatives. The problem with all of this is that there are IVP’s out there in the world that have initial values at places other than t = 0 t = 0.
How do you decompose IVP’s?
First let’s get the partial fraction decomposition. Now, plug these into the decomposition, complete the square on the denominator of the second term and then fix up the numerators for the inverse transform process. Finally, take the inverse transform. To this point we’ve only looked at IVP’s in which the initial values were at t =0 t = 0.