Solve an integro differential equation?

Questions
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1

Hi all,

Wolfram language has the capability of solving an integro differential equation as described in the following article.

However, I was trying to use this to solve an integro differential equation of the form but couldn’t get a solution. Can somebody help me understand why and tell me a how I could solve a similar equation please?

Question.PNG

2

Use DSolve, like said article.

3

@Tonio JimenezC

Tried with that. But it doesn’t give me a solution instead it returns the initial statement.

4

Please send details on what you did exactly.

5

I’ve attached the script herewith. I simply followed the steps in the link above.
Thanks in advance.

Question.nb (3.73 KB)

6

Unfortunately,DSolve[]'s support for integral equations is still somewhat limited,
so don’t be surprised if some things don’t work yet.

We can (solve ?) these equation by Laplace Transfrom ,but Mathematica can’t find Inverse Laplace Transfrom.

 eqn = Derivative[1][y][t] == -(c^2*Integrate[E^(-(b*(t - \[Tau])) - (a^2*(t - \[Tau])^2)/4)*y[\[Tau]], {\[Tau], 0, t}])
 sol = LaplaceTransform[eqn, t, s]
 sol2 = Solve[sol, LaplaceTransform[y[t], t, s]]
 y[t] == InverseLaplaceTransform[LaplaceTransform[y[t], t, s] /. sol2[[1]], s, t]

 (*y[t] == Sqrt[a^2]
    InverseLaplaceTransform[1/(
    Sqrt[a^2] s + 
     c^2 E^((b + s)^2/a^2) Sqrt[\[Pi]] Erfc[(b + s)/Sqrt[a^2]]), s, 
    t] y[0]*)

Regards M.I.

7

Yes. I guess so. Tried the Laplace transform approach also and ended up having the same issue.
Waiting for a feedback from the Walfram team.