So I'm trying to solve this partially decoupled system:

$\displaystyle \frac{dx}{dt} = 2x + 3y$

$\displaystyle \frac{dy}{dt} = -4y$

1.) Solve it WITHOUT using matrices. (HINT: start with the second equation)

2.) Solve it with using matrices.

I can do #2... but #1 is so hard.

For #2, I set up the system into a matrix so I get X' = AX, with A being:

[[2,3],[0,-4]]

Then, I found the eigenvalues:

$\displaystyle \lambda_1 = 2, \lambda_2 = -4$

And the corresponding eigenvectors (call them $\displaystyle k_1, k_2$, respectively:

k_1 = [[1],[0]], k_2 = [[1/2],[1]

So then the solution is:

$\displaystyle X = c_1e^{2t}k_1 + c_2e^{-4t}k_2$

Voila.

Now for #1...

So solving the 2nd equation first...:

$\displaystyle \frac{dy}{-4y} = dt \implies \frac{-ln(|y|)}{4} = t + C$

Solve for y (I ignored the absolute value so I don't get 2 diff soln's):

$\displaystyle y = e^{-4t-4c}$

So now plug this into 1st eq.:

$\displaystyle \frac{dx}{dt} = 2x + 3\cdot e^{-4t-4c}$

No idea how to solve this! Maybe I did something wrong.. no idea.