Question

I have to give the general solution of the DEs system. Can someone please check my solution?

Answer 1

\[x ^{'}=\left[\begin{matrix}4 & -3 \\ 6 & -7\end{matrix}\right]x\]

equation to find the eigenvalues:
\[r ^{2}+3r-10=0\]
eigenvalues are {-5,2}

eigenvectors for r=-5: (1,3)
eigenvectors for r=2: (3,2)

general solution
\[x(t)=c _{1}\left(\begin{matrix}1 \\ 3\end{matrix}\right)e ^{-5t}+c _{2}\left(\begin{matrix}3 \\ 2\end{matrix}\right)e ^{2t}\]

Answer 2

Perfectly done. I guess I might as well share a trick I use to compute eigenvectors: \[\Large \left[\begin{matrix}4 & -3 \\ ... & ...\end{matrix}\right]\left(\begin{matrix}a \\ b\end{matrix}\right)=\lambda \left(\begin{matrix}a \\ b\end{matrix}\right)\]

I just solve for my eigenvector once by looking at the top equation we get:

\[\Large 4a-3b=\lambda a\]Solve for b and plug this in: \[\LARGE a\left(\begin{matrix}1 \\ \frac{4-\lambda}{3}\end{matrix}\right)\]

I usually see people do some really convoluted thing where they try to plug it back in and subtract from the diagonal for each eigenvalue or something terrible, so I thought I'd help possibly!

Answer 3

Very nice trick to know! Yeah, I did the convulated thing, hehehe

Answer 4

Also, I suppose I should tell you of an easy way of checking yourself. Just plug this back into the differential equation to see if it satisfies it. Do you know how to do that for matrix equations like this?

Answer 5

Well in the first part of the question I was given 2 solutions and had to check whether they actually were solution to the system. And I learned to do this (tonight btw) by first multiplying the matrix with the given x solution and thus have x'. And then I took the derivative of the given solution and checked whether that would give the same result as the calculated x'. I guess I could do the same thing in general.

Answer 6

Yeah, it's definitely one of the fastest ways to check yourself as well as get faster at multiplying matrices in your head if possible.

It's about as reliable as differentiating your answer after solving an integral to see if you did it right.

Answer 7

Next part of the question will be finding a particular solution with given starting values, but I need to work through the theory of that first and since it's late, I'll do that tomorrow.

Answer 8

Ok good luck, if I get on tomorrow I'll be on the lookout. =P