Variation of parameters, Mathematics

In this case we will require deriving a new formula for variation of parameters for systems.  The derivation now will be much simpler than the when we first noticed variation of parameters.

First assume that X(t) be a matrix whose ith column is the ith linearly independent solution to the system,

x?' = A x?

Now this can be illustrated that X(t) will be a solution to the subsequent differential equation.

X' = AX    ......................(1)

It is nothing more than the original system along with the matrix instead of the original vector.

We are going to attempt and get a particular solution to,

x?' = A x? + g? (t)

We will suppose that we can get a solution of the form,

x?p = X(t) v? (t)      

Here we will need to find out the vector v? (t). To do it we will require plugging this in the nonhomogeneous system. Keep in mind to product rule the particular solution whiler plugging the guess in the system.

X' v? + X v? = AX v? + g?

See that we dropped the "(t)" part of things to identify the notation a little. Here by using (1) we can rewrite this a little.

X' v? + X v? = X' v? + g?

X v? = g?

Since we formed X using linearly independent solutions we identify that det(X) should be nonzero and this in turn implies that we can get the inverse of X. Therefore, multiply both sides with the inverse of X.

v? = X-1 g?

This time all that we require to do is integrate both sides to get v? (t).

v? (t) = ∫ (X-1 g?) dt

When with the second order differential equation case we can ignore any constants of integration. So, the particular solution is,

x?p = X ∫ (X-1 v? (t)) dt

Posted Date: 4/11/2013 1:50:30 AM | Location : United States

Related Discussions:- Variation of parameters, Assignment Help, Ask Question on Variation of parameters, Get Answer, Expert's Help, Variation of parameters Discussions

Write discussion on Variation of parameters
Your posts are moderated
Related Questions
The sum of two consecutive integers is 41. What are the integers? Two consecutive integers are numbers in sequence like 4 and 5 or -30 and -29, that are each 1 number apart. Le

Show the result of the following sequence of UNION operations using union-by-weight with the following assumptions Unions are performed on the representatives on the sets th

Explain Expressions ? "One set of absolute value signs can only take the absolute value of one number." For example, For the absolute value of negative six plus three,

Evaluate following limits. Solution Therefore we will taking a look at a couple of one-sided limits in addition to the normal limit here. In all three cases notice

Diffrent type of rectillinar figure..

Solution of quadratic equations, please provide me the assignment help for solving the quadratic equations.

2, -8, 32, -128, ?, ?, ?, what are these next 3?

what is transportation and assignment problem. give the computer application of transportation and assignment problem

In the National Hockey championship, there are 30 independent ice hockey teams. Every of the teams will play 82 official NHL games every year. Many teams will have to travel from t

4. Two hosts, one on East (host A) and one on the west coast (host B) of the USA are exchanging data. Suppose A is sending a large file to B. The file is split into packets of size