Nonhomogeneous system particular solution.

[Pavoo]

Homework Statement

Verify that the vector functions $x_{1}=\begin{bmatrix}e^{t}\\ e^{t}\end{bmatrix}$ and $x_{2}=\begin{bmatrix}e^{-t}\\ 3e^{-t}\end{bmatrix}$ are solutions to the homogeneous system

$x'=Ax=\begin{bmatrix}2 & -1 \\ 3 & -2 \end{bmatrix}$ on $(-\infty ,\infty )$

and that

$x_{p} = \frac{3}{2}\begin{bmatrix}te^{t}\\ te^{t}\end{bmatrix}-\frac{1}{4}\begin{bmatrix}e^{t}\\3e^{t}\end{bmatrix} + \begin{bmatrix}t\\2t\end{bmatrix}-\begin{bmatrix}0\\1\end{bmatrix}$ is a particular solution to the nonhomogeneous system x'=Ax + f(t), where f(t)=col($e^{t},t$).

Find a general solution to x' = Ax + f(t).

Homework Equations

I can handle the first part, basically showing with a Wronskian wether the solutions form a fundamental solution set.

The second part, about the particular solution - completely stuck. I understand that the general solution is written in the form x = xp +Xc, so that is no problem.

The Attempt at a Solution

Should I use eigenvectors, or should consider another way, such as described here?

http://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousSystems.aspx

Really thankful for all the hints here because the chapter in the book does not even cover particular solutions.

So, how do I verify that that particular solution is correct (meaning how do I derive a particular solution from the above data)?

 

[SteamKing]

IMO, all that is required is that you verify (i.e., plug and chug) that xp satisfies the inhomogeneous system.
If it does, then write the general solution using xp and xc. I don't think you are expected to derive xp from scratch.

 

[Mark44]

IMO, all that is required is that you verify (i.e., plug and chug) that xp satisfies the inhomogeneous system.

Also, verify that x₁(t) and x₂(t) are solutions of the homogeneous system.

If it does, then write the general solution using xp and xc. I don't think you are expected to derive xp from scratch.

Right. Problems like this are relatively simple. You're given a potential solution and are asked to confirm that it is indeed a solution. Substitute the given functions into the appropriate differential equation and you should end up with a statement that is identically true.

 

[Pavoo]

Right. Problems like this are relatively simple. You're given a potential solution and are asked to confirm that it is indeed a solution. Substitute the given functions into the appropriate differential equation and you should end up with a statement that is identically true.

Thanks for the advice but I'm sorry - this is not enough for me to understand. I need to find out the way to derive it, because I am certain that I will get this on the coming test.

Here's what I found out on the them internets - but I got stuck here as well. I understand the general way to solving it, but maybe it's just that I'm stuck in some constants. Looking at the particular solution, this can be defines as:

$x_{p}=ate^{t}+be^{t}+ct+d$

Looking at the formulas above, this can now be rewritten as:

$ate^{t}+(a+b)e^{t}+c=Aate^{t}+Abe^{t}+Act+Ad+ \begin{bmatrix}1 \\ 0 \end{bmatrix}e^{t}+ \begin{bmatrix} 0 \\ 1 \end{bmatrix}t$

So now I have: $Aa=a, Ab=a+b-\begin{bmatrix}1\\0\end{bmatrix},Ac=-\begin{bmatrix}0\\1\end{bmatrix}, Ad=c$

Where A all along is $A=\begin{bmatrix}2 & -1\\ 3 & -2\end{bmatrix}$

I get the constant $a=\frac{3}{2}$ which is correct. However, I am completely stuck with the constant b. I am calculating using $\begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}$. But I get two exactly same equations.

Thank you for your help on this one!