Matrix methods for systems of differential equations

I show how to use matrix methods to solve first order systems of differential equations. The ideas involve diagonalization and basic linear ODEs.
The example shown involves symmetric matrices.

[DrChrisTisdell]

Yes, examples are important, but why things work is also important. I aim balance between theory, methods and applications..

[DrChrisTisdell]

@TheXiwi For a defective matrix (ie, a square (n x n) matrix that does not have n linearly independent e.vectors and, hence, is not diagonalizable) the procedure is to form a complete basis by augmenting the eigenvectors with generalized eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems. (See the wiki page on "Defective matrix")
However - this is beyond the scope of this presentation on symmetric matrices.

[DrChrisTisdell]

@MarlosZappa OK, so let's say we have a 2x2 system and h_1 is zero (but h_2 isn't). Our system is then y_1' = \lambda_1 y_1 and y_2' = \lambda_2 y_2 + h_2(t). These can be solved individually, there is no need to worry about particular solutions.

[DrChrisTisdell]

Hi - from the matrix equation at the bottom of the page: 1v_1 + 1v_2 = 0, ie v_1 = -v_2.

[DrChrisTisdell]

Hi - the P matrix must have columns each of length 1, so we normalized our eigenvectors by multiplying by multiplying by 1/\sqrt{2}.

[ranganatha.b.r]

Beneficial video. I was able to understand, even though I knew nothing about it. Thanks a lot

[MarlosZappa]

@DrChrisTisdell Well the problems we were given go up to 4x4 =P 1st I find the solution to the homogeneous system, then I go ahead to the whole transformation business to find particular solutions. The question is: what to do when one of the transformed solutions is general, because one of the transformed equations is homogeneous. When you use those particular solutions to transform the system back, the one to the homogeneous equation is the general solution with an arbitrarily chosen K?

[niyaziadamsiken]

I recommend the e^t*A method with inhomogenous systems

[DrChrisTisdell]

Let me ask you this: what values satisfy v_1 = v_2? There are infinitely many. So tet's choose the simplest ones: take v_1 = 1 and so v_2 = 1. That's how we got the VECTOR v_1. Now, consider the equation v_1 = -v_2 (note that they are different from the v_1 and v_2 that we calculated above)
. There are infintiely many solutions, so let's choose a simple case, choose v_1 = 1 and so v_2 = -1. That's how we get the VECTOR v_2.

[nilufargh]

Hey 
Thanks for the video, it was great! 
everything makes sense, but I just have one question, why did you multiply  1/sqrt(2) to  p matrix??

[TheLastXCloud]

Thank you, a thousand times, you really saved me before the exam

[dashz]

apart from the 1/sqrt2, everything else is perfectly explained. thank you very much sir.
would you mind explaining the 1/sqrt2 a bit more? u replied to a guys that 'the P matrix must have columns each of length 1, so we normalized our eigenvectors by multiplying by multiplying by 1/\sqrt{2}. ' but i dont quite get it. thank you again.

[juandiegodurango]

Hello.
First of all, sorry by my english; in this moment it's not very good yet.... I'll try to be as clear as possible.
1. Can you give me a link of your channel (or any other) in which an ODE system of higher order is solved by a similar method of this video, please?
2. Can you give me any reference?
3. Is there a problem of real life that you know that is modeled with an ODE system?

I'll really appreciate you any answer that you can give me, and of course thank you for this video.
¡Muchas gracias!

[tomj]

really appreciate the video, is there anywhere i can find the notes used? i know you have them available with other videos

thank you!

[MarlosZappa]

@DrChrisTisdell Yeah, that's it, but I meant actually when you're solving the transformed system P^(-1)*A*Py + P^(-1)g. You might have one of the lines of g as a 0. Since you're looking for particular solutions, and you get exponential solutions with general constants in that case, how do you get a valid particular solution useful for this method? Can you use 0, or do you have to use the exponential solution with, say, 1 as a constant, as a particular solution?

[DrChrisTisdell]

@MarlosZappa Sorry, I'm finding it hard to fully understand your question. Are you meaning the case when all the h_i functions are zero at 04:45? If so then then you have the system y' = Dy which will give rise to exponential solutions.

[lecolis]

which playlist is this video on in your channel? or which playlist include videos of the course in which this video was teaching to?

[donskanone]

Thanks a lot for sharing all your knowledge. Your videos are very helpful.

But what about the following Problem:
x'(t)=A*x(t) where A=[0, B, 0;-B, 0, 0;0, 0, 0] and x is a Vector in R3.

The eigenvalues of A are zero. I think x has to deal with sines and cosines but I don't like to guess and don't know how to derive the general solution to this problem. Can anyone help me?

[MarlosZappa]

Thank you very much! One question: if you have one of the y-equations as not being inhomogeneous, is the particular solution 0, or is it the homogeneous solution with 1 as a constant? Or can you set the constant as 0 and say 0 is a particular solution?

[britfreeman]

Will this method work if the matrix is not symmetrical? I am trying to figure out how to solve problems like this, but the matrices are not symmetric.