I want to solve the following system of differential equations: $$\begin{cases} x'(t) = x(t) + y(t) \\ y'(t) = y(t) \end{cases}$$
With the intial conditions $x(0) = 1$ and $y(0) = 1.$
Obviously, $y(t)$ (as well as $y'(t)$) are equal to $e^t$.
How do I find $x(t)$ now?
I have tried the following:
$x'(t) = x(t) + y(t)$ can be written as
$y(t) = x'(t) - x(t).$
Now we know that $x(0)$ and $y(0) = 1$, hence: $1 = x'(0) - 1, \therefore x'(0) = 2.$
How do I proceed from here (if it makes sense to proceed from here)?