Differential equations with matrices and eigenvalues?

[stefan1988]

Homework Statement

this is the homework that i have to do
http://img690.imageshack.us/img690/2783/problemsb.png

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The Attempt at a Solution

im not really sure if this is the right method but i will solve it like if it was a homogeneous equation by setting the right side equal to 0
then one i have that i would solve for roots
once that i would solve for constants and get final equation to be something as y=c1e^x1+c2e^x2

but i don't think that sounds right because I am not really finding rt) and last part professor hasn't really covered how to do this sort of thing with matrices i know how to solve for eigenvalues but i don't know how to get it to that point or how do you go about working out that problem

if only i know what i should be studying for or looking for i would be able to solve it but i been looking everywhere and can't find anything to help me out with it

i appreciate any help =D

 

[mighty2000]

Hello,

Based on the information provided in the image, it seems like you have a system of linear differential equations. In order to solve this type of problem, you will need to use methods such as matrix diagonalization or eigenvalues and eigenvectors.

First, you will need to rewrite the system in matrix form, where the coefficients of the dependent variables (y1 and y2) are the entries of the matrix. In this case, the matrix will be a 2x2 matrix.

Next, you will need to find the eigenvalues and eigenvectors of this matrix. This can be done by finding the roots of the characteristic equation and solving for the corresponding eigenvectors.

Once you have the eigenvalues and eigenvectors, you can use them to write the general solution of the system. This will involve using the exponential function, as you mentioned in your attempt at a solution.

I would recommend reviewing your notes or textbook for more information on how to solve systems of linear differential equations using eigenvalues and eigenvectors. Good luck with your homework!