Lecture Notes for Exam 3

Lecture Notes on Material for Exam III--Not all the theory is here but it will give you an idea of what was covered in class.

Given that we are somehow able to find the complimentary solution (solution to the corresponding homogeneous equation) to an equation of the form given below, we want to find a particular solution in cases where our previous annihilator method using undetermined coefficients cannot be applied to produce a form for y_p.

(1)

Divide each side of (1) by to get

(2) .

Rewrite this as

(3) .

Suppose we somehow find that where is the general solution to

(4) .

It turns out to be possible (subject to being able to do the required integration) to find a of the form

.

We need to substitute this form for into (3) and try to figure out what A(x) and B(x) (both potentially functions of x) need to be. To do this we need to find and .

To avoid dealing with the second derivatives of A(x) and B(x) we will look for A and B satisfying the following condition:

Thus

Substituting into (3) yields

(5)

Since is a solution to (4) the terms in red above must add up to 0. Similarly, since is a solution to (4) the terms in blue must add up to 0. Thus (5) becomes

.

To find A and B we need to solve the system

Cramer’s Rule produces

As an alternative you could use the reduction in order option for finding y_p.  This formula often (but not always) leads to more complex integration.

The variation of parameters formula can be extended to find particular solutions to linear differential equations of order higher than two but you will not be expected to do so in this class.

Example

It is possible to find y_p through an integration formula for a first order linear differential equation in a manner similar to what you see demonstrated above as follows:

Laplace Transform

The Laplace Transform is an integral transform that transforms a function of some independent variable, say t, into a function of s.  The function g(s,t) is called the kernel function for the Laplace Transform.

The Laplace Transform of some additional functions can be found without the need to evaluate improper integrals by differentiating transforms that have already been established.

We can also pretty quickly find the Laplace Transform of functions of the form e^atf(t) if we already know the Laplace Transform of f(t).

To use the Laplace Transform to solve a differential equation we are going to need to be able to represent the Laplace Transform of the derivative of a function in terms of the Laplace Transform of the original function under the assumption that the Laplace Transform of the original function exists (i.e., the limit that defines the improper integral that defines the Laplace Transform exists).

To use the Laplace Transform to solve an initial value problem, we need to take the Laplace Transform of each side of the equation, implement the initial conditions, algebraically solve for the Laplace Transform of the solution function we are looking for, L{y}, and then use the Inverse Laplace Transform, L^-1, to find y.

more examples

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

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