What do you do to separate the X terms from the y terms in an equation?
Example 1
- First we separate all of the y terms from the x terms by putting each type of term in a different side of the equal sign:
- We integrate each side, the side containing the y terms must be integrated with respect to y, and the side containing the x terms must be integrated with respect to x.
Does every ordinary differential equation have a solution?
Not all differential equations will have solutions so it’s useful to know ahead of time if there is a solution or not. This question is usually called the existence question in a differential equations course.
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.
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 do differential equations have to be solved simultaneously?
This will lead to two differential equations that must be solved simultaneously in order to determine the population of the prey and the predator. The whole point of this is to notice that systems of differential equations can arise quite easily from naturally occurring situations.
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.