decoupling and integrating differential equations I am having trouble with the process of decoupling. If I have
$$\frac{dx}{dt}=-x+y$$
$$\frac{dy}{dt}=-x-y$$
I am trying to figure out how to solve for $x(t)$ and $y(t)$ by decoupling the system so that I only have one variable but I can't seem to get anywhere

Note that you can write your system like so:
$$
   \frac{\mathrm{d}}{\mathrm{d}t}\begin{pmatrix}x\\y \end{pmatrix}
   +\begin{pmatrix}1 & -1 \\ 1 & 1\end{pmatrix}\begin{pmatrix}x\\y \end{pmatrix}
   =0.
$$
To effectively decouple the equations, you want the system to looke like
$$
   \frac{\mathrm{d}}{\mathrm{d}t}\begin{pmatrix}u\\v \end{pmatrix}
   +\begin{pmatrix}a & 0 \\ 0 & b\end{pmatrix}\begin{pmatrix}u\\v \end{pmatrix}
   =0,
$$
so you need to diagonalize the matrix $\begin{pmatrix}1 & -1 \\ 1 & 1\end{pmatrix}$, find the eigenvalues $a$ and $b$, and the corresponding eigenvectors $u$ and $v$. In this case, we find $a=1+i$, $b=1-i$, $u=(1,-i)$ and $v=(1,i)$. The final system is thus
\begin{align}
    \frac{\mathrm{d}}{\mathrm{d}t}u+(1+i)u=0 \\
    \frac{\mathrm{d}}{\mathrm{d}t}v+(1-i)v=0
\end{align}
with $u=x-iy$ and $v=x+iy$.
This is a somewhat more general method, although probably slightly not as effective for this particular problem, then the one T. Bongers suggested.
