# Thread: general solution for differential equations

1. ## general solution for differential equations

Hi I wasn't sure whether to post this in the linear algebra forum or the calculus forum, but it deals with differential equations. I'm having a hard time understanding this one question and how it differs from the normal case. The question is:

Find the general real-valued solution for the system of equations:

$y'_1 = 3y_1 + y_2$

$y'_2 = y_1 + 3y_2$

Now, I know how to solve equations of the form

$y' = 3y + f$

where f is just some function or constant. The confusing part for this problem is that $y_1$ & $y_2$ are in both equations, so you can't just solve each equation outright like you typically would when there is just y and not the y1 and y2.

Once the general equation is known, I know its pretty easy to solve the initial value problem I need to solve after, but I just need help on how to approach the problem.

Am I supposed to somehow substitute to get one equation with all y1 and the other with all y2 or how would I go about doing it? It just seems like a weird problem to me that I've never encountered before.

2. We could write it as:

$x'=2x+y$

$y'=x+3y$

Now, we can find the eigenvalues:

$\begin{bmatrix}3-{\lambda}&1\\1&3-{\lambda}\end{bmatrix}$

${\lambda}^{2}-6{\lambda}+8=0$

${\lambda}=2, \;\ {\lambda}=4$

Now, can you proceed?.

Start by plugging the eigenvalues we just found back in for lambda in the matrix above.

3. Yes, thanks that makes A LOT more sense lol, thank you very much.