# Solving coupled integro-differential equations using Laplace transform

I have two coupled integrodifferential equations, as shown in the image attached.

I am trying to solve them using a Laplace transform, as follows

\[Alpha]cont[\[Tau]_] := (E^(I (\[Tau]) \[Omega]0) gab^2 \[Pi]^(3/2) r0^2 \[Rho]0)/(Sqrt[2] \[HBar]^2 (r0^2 + (I (\[Tau]) \[HBar])/mb)^(3/2))
\[Alpha]s = LaplaceTransform[\[Alpha]cont[\[Tau]], \[Tau], s];
sols1 = c1s /. Solve[s c1s - 1 == -\[Alpha]s (c1s + c2s), c1s]
sols2 = c2s /. Solve[s c2s - 0 == -\[Alpha]s (c1s + c2s), c2s]


so that I can then evaluate the functions using the Stehfest algorithm. However, I am unable to produce this.

• What issue have you encountered exactly? Does the code return any errors? If so, which ones? If not, where does it stop working? – MarcoB Mar 16 at 13:03
• If you probably mean ? :$$\text{c1}'(t)=-\int_0^t (\text{c1}(\tau )+\text{c2}(\tau )) \text{\alpha cont}(\tau ) \, d\tau$$ then LaplaceTransform in this case, nothing will do. – Mariusz Iwaniuk Mar 16 at 14:11