How Do I Solve This Initial Value Problem Using a Given Solution Function?

[BlkDaemon]

I'm hacking at this particular linear system:

dY/dt = [1, -1; 1, 3] Y

I've already found myself a solution using the following function:

Y(t) = [ te^(2t), -(t+1)e^(2t) ]

That was fun, actually, once I figured out what the hell I was doing.

Here's my question: the next part of the problem asks for a solution to the intial value problem of the above differential equation, where

Y(0) = [0, 2]

I'm not sure how to attack the problem. Do I sub that matrix in for both values of Y, on both sides? If I do, then where do I put Y(t) so that this works out correctly? Or is there some other matrix multiplication step that I'm missing?

Any ideas?

Thanks in advance.

BlackDaemon

 

[Tide]

You have a solution for the system. Generally, you will have a superpostion of two linearly independent solutions and you will adjust their coefficients to match the initial values.

 

[wais]

First of all, great job on finding a solution to the given linear system! Now, to solve the initial value problem, you can simply plug in the values of Y(0) = [0, 2] into your solution function Y(t) = [te^(2t), -(t+1)e^(2t)]. This will give you the specific values of Y at time t=0. So, your solution to the initial value problem would be Y(t) = [0, 2e^(2t)]. This means that at time t=0, the values of Y are Y(0) = [0, 2], just as given in the problem.

You do not need to do any matrix multiplication or substitution, as your solution function already takes into account the values of Y at different times. Just plug in the values and you will have your solution to the initial value problem.

I hope this helps clarify the process. Keep up the good work!