Initial value problem with a delta term

Question

Im having trouble solving this initial value problem. I know how to solve it without the delta-term (C1*e^(lambda*t)*S1 + C2*e^(lambda*t)*S2), but how do i solve it with a delta term?

Answer 1

Hint: for $t<1$, the delta function term is zero, and the set of equations reduces to the system you already know how to solve. Likewise on the domain $1<t$. Find the general solution on either side of $t=1$, then apply the proper matching conditions at $t=1$ to determine the remaining undetermined coefficients. (Note that the first derivative will have a jump discontinuity at this point.)

Answer 2

@AhmedD. Here's another approach using Laplace transform. The basic idea is to take $\mathcal{L}_s$ of both equations and solve for $X(s)$ and $Y(s)$ so we can obtain ultimately $x(t)$ and $y(t)$ anti-transforming.

Here's the code I have used to compute the solution using $\texttt{Mathematica}$.

system  = {x'[t] == 3 x[t] - y[t] + DiracDelta[t - 1], y'[t] == x[t] + y[t]};
systems = LaplaceTransform[system, t, s] /. {x[0] -> 1, y[0] -> 1};

sollist = Solve[systems, {LaplaceTransform[x[t], t, s], LaplaceTransform[y[t], t, s]}];

xt = InverseLaplaceTransform[LaplaceTransform[x[t], t, s]/.sollist[[1]][[1]], s, t];    
yt = InverseLaplaceTransform[LaplaceTransform[y[t], t, s]/.sollist[[1]][[2]], s, t];

Plot[{xt, yt}, {t, 0, 1.25}, AxesLabel -> Automatic]

And here is the result of the plot where you can see the jump of $x(t)$ (blue solid) at $t=1$: