I am being asked to solve a system of equations with initial conditions, however, there is a dirac delta function involved in one of the equations. Would the answer(s) include the dirac delta function or would there be another way of writing it?
For reference, here is the question:
$$x'+3y=\delta\left(t-7\right),\quad \;\ y\left(0\right)=0$$ $$x-y'=0,\qquad \qquad \quad\ x\left(0\right)=0$$
For $t \lt 7$ the right sides are zero, so $x'=y'=0$ and $x=y=0$. If we integrate across $t=7,$ the integral of the right side of the first equation jumps from $0$ to $1$, so the left side must, too. This gives us $\lim_{t=7^+}x=1$. We can then solve the equations starting from $t=7$ with initial conditions $x(7)=1,y(7)=0.$