Finding General Solution / Fundamental matrix

• donkeystalk
In summary, the conversation is about a student studying for a test and having trouble with a practice question involving a matrix equation. The given answer is Y(t) = [1/2(e^t - e^-t), e^-t(t+1), 0]. The student is looking for the steps to reach this solution and has searched for help online, but with no luck. The recommended method is to search for the matrix exponential and state equation in a linear systems book, as the solution involves a convolution integral.
donkeystalk
Morning everyone,

Studying for a test and having a problem on a practice question he gave us to study with. Here's the question along with the answer:

Y' = AY + [e^t
e^-t
0]

with A =
[-1 0 4
-0 -1 2
0 0 1]

the answer given is: Y(t) =
[1/2(e^t - e^-t)
e^-t(t+1)
0]

My question is, what are the steps to getting to this solution, I've gone over notes, examples, scavenged the internet, not a whole lot of luck. I know it's probably out there, but I am probably looking up the wrong keywords via Google.

Any help with a general outline of what to do would be extremely helpful! Thanks a lot!

You can search for the matrix exponential, and state equation in any linear systems book.

$$\dot x = Ax+Bu$$

is the form that you are looking for. The solution is a not-so-complicated convolution integral

trambolin said:
You can search for the matrix exponential, and state equation in any linear systems book.

$$\dot x = Ax+Bu$$

is the form that you are looking for. The solution is a not-so-complicated convolution integral

ok so what I am looking for is the state equation and matrix exponential?

this type of problem ended up not being on the exam,but I am guessing it will show up on the final in a just over a week.

1. What is a general solution/fundamental matrix?

A general solution or fundamental matrix is a mathematical tool used in linear systems of differential equations to find a set of solutions that satisfy the given initial conditions. It is represented as a matrix containing all possible solutions to the system.

2. How is a general solution/fundamental matrix different from a particular solution?

A particular solution is a single solution to a specific set of initial conditions, while a general solution or fundamental matrix contains all possible solutions for a given system. A particular solution can be derived from a general solution by substituting the initial conditions into the matrix.

3. What is the importance of finding a general solution/fundamental matrix?

Finding a general solution or fundamental matrix is crucial in solving linear systems of differential equations, as it provides a complete solution set that can be used to find any specific solution for the given system. It also helps in understanding the behavior and properties of the system.

4. What are the steps involved in finding a general solution/fundamental matrix?

The steps for finding a general solution or fundamental matrix include setting up the system of differential equations, finding the eigenvalues and eigenvectors of the coefficient matrix, creating a diagonal matrix with the eigenvalues, and using the eigenvectors and diagonal matrix to construct the general solution or fundamental matrix.

5. Are there any limitations to using a general solution/fundamental matrix?

One limitation of using a general solution or fundamental matrix is that it can only be applied to linear systems of differential equations. It also requires the system to have constant coefficients and be in a standard form. Additionally, the general solution may not always be unique, and further analysis may be needed to find a particular solution.

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