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This question already has an answer here:

enter image description hereenter image description here

I do not know how to handle the integro-term.

ieqn = 
 1 - 6.25*10^5*x[t] + 1.234*10^4*Integrate[x'[u]/Sqrt[t - u], {u, 0, t}] == 1.5924*x''[t];
ic = {x[0] == 0, x'[0] == 0};
sol = DSolve[{ieqn, ic}, x[t], t];
Plot[x[t] /. sol, {t, 0, 0.007}]

enter image description here enter image description here

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marked as duplicate by xzczd, MarcoB, AccidentalFourierTransform, halirutan Jul 10 '18 at 23:10

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • 2
    $\begingroup$ Is this about Wolfram Mathematica? Have you tried do find something in integro-differential topics? $\endgroup$ – Kuba Jul 3 '18 at 12:39
  • $\begingroup$ the value of t do not change the output of the figure or curve,and i do not know what's wrong with my code $\endgroup$ – Hukai Jul 4 '18 at 1:26
  • $\begingroup$ Which version are you in? In v11.2 DSolve returns unevaluated. $\endgroup$ – xzczd Jul 5 '18 at 10:38
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I use "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)".

Copy-pasting the proposed code I receive only a slew of errors and no output (only the axes, but no graph).

Instead, writing:

f[u_?NumericQ] := x'[u]/Sqrt[t - u]
ieqn = 1 - 6.25 10^5 x[t] + 1.2341 10^4 Integrate[f[u], {u, 0, t}] == 1.5924 x''[t];
ic = {x[0] == 0, x'[0] == 0};
xsol = NDSolveValue[{ieqn, ic}, x, {t, 0, 0.01}] // Quiet;
Plot[xsol[t], {t, 0, 0.01}] // Quiet

I get:

enter image description here

where the two Quiet still hide a series of errors.

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  • $\begingroup$ I'm afraid this solution is not reliable, because you've hidden the x'[u] in a blackbox function, NDSolve won't be able to handle it corectly. $\endgroup$ – xzczd Jul 5 '18 at 10:26

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