# I need to solve integro–differential equation with derivative

There are 2 equations a1 and a2, the difference between them is the red words part(integral term). Boundary x(0)=0,x'(0)=0

x'(t-u) means Bring t-u into the derivative of x ，for example, if x(t)=t^3, then x'(t-u)=3*(t-u)^2

However ,I get the same curve by different codes .so I want to know what's wrong with my code especially the integral term(red words).

The following is code of a1

ieqn = 1 - 6.25*10^5*x[t] +
1.234*10^4*Integrate[x'[t - u]/Sqrt[u], {u, 0, t}] == 1.5924*x''[t];
ic = {x == 0, x' == 0};
sol = DSolve[{ieqn, ic}, x[t], t];
Plot[x[t] /. sol, {t, 0, 0.005}]


The following is code of a2

ieqn = 1 - 6.25*10^5*x[t] == 1.59236*x''[t];
ic = {x == 0, x' == 0};
sol = DSolve[{ieqn, ic}, x[t], t];
Plot[x[t] /. sol, {t, 0, 0.005}] This equation is from this paper

The following is the correct curve from this paper.

correct curve of a1:red curve arrowhead

correct curve of a2:dashed black arrowhead

As far as I know, those equation can only have numerical solution. In the following website, they discussed the same question with mine, but I think there is some wrong.

At last ,I wonder what's wrong with my code especially the integral term(red words).

• The units in my equations is m and s,and those in figure of paper is um and ms,so the Magnitude of my output is correct Jul 10, 2018 at 3:00
• What version of Mathematica are you using? Jul 10, 2018 at 5:56
• Your equation is essentially the same as the one in the linked post. Have you tried the method based on Laplace transform in my answer? Jul 10, 2018 at 7:10
• If you are using version 11.0, you are a victim of the bug described in 134465. DSolve instead returns unevaluated in later versions instead of just ignoring the integral. Jul 10, 2018 at 13:20