# How to solve ode with convolution?

my question is as this and i want to get the solution of this ode x(t)

$$({M_{33}}{\rm{ + }}{{\rm{m}}_{33}})x''(t) + \int\limits_{ - \infty }^t {x'(tau){K_{33}}(t - tau)d{\rm{ }}tau} = {F_3}(t)\\ {K_{33}}(t) = \frac{2}{\pi }\int\limits_0^{ + \infty } {{b_{33}}} (\omega )\cos (\omega t)d\omega \\$$

and the defined value is as this

$${F_3}(t)= 31.4380978232711cos(3.9893240045584673t);\\ M_{33} = 1.95;\\ m_{33} = 2.5;$$

codes asre as this

b33 = 2.52077405993092/(46.4663900509659 + omega^2 -
11.8316929905336*omega);
k33 = 2/\[Pi]*Integrate[b33*Cos[omega*t], {omega, 0, +\[Infinity]}]

(*1.60477 If[
t \[Element]
Reals, (0.0190724 -
0.0333163 I) ((3.33766 -
1.91069 I) Cosh[(3.38661 + 5.91585 I) Abs[
t]] CosIntegral[(-5.91585 + 3.38661 I) Abs[
t]] - (0.955346 +
1.66883 I) Sinh[(3.38661 + 5.91585 I) Abs[t]] (\[Pi] -
2. SinIntegral[(-5.91585 + 3.38661 I) Abs[
t]])) + (0.0190724 +
0.0333163 I) ((3.33766 +
1.91069 I) Cos[(5.91585 + 3.38661 I) Abs[
t]] CosIntegral[(-5.91585 - 3.38661 I) Abs[t]] + (1.66883 +
0.955346 I) Sin[(5.91585 + 3.38661 I) Abs[t]] (\[Pi] +
2 SinIntegral[(5.91585 + 3.38661 I) Abs[t]])),
Integrate[Cos[omega t]/(
46.4664 - 11.8317 omega + omega^2), {omega, 0, \[Infinity]},
Assumptions -> t \[NotElement] Reals]]*)


int2 = Integrate[x'[tau]*k33[t - tau], {tau, -[Infinity], t}]

it is x'[tau] rather than x'[t]

ft = 31.4380978232711 Cos[3.9893240045584673 t];
M33 = 1.95;
m33 = 2.5;

equation = (M33 + m33)*x''[t] + int2 == ft;
ts = 50;

s1 = NDSolve[{equation, x == 0, x' == 0}, x, {t, 0, ts}]
Plot[Evaluate[y[x] /. s], {x, 0, 30}, PlotRange -> All]


however it doesn't work so how to solve it?

updated3

now i use fit method to get the b33 and we can use this b33 to get the k33 but it seems that this k33 figure is queer from t=0-100.

k33 in some essay is as this updated4

now the question changes to this ,how to solve the integral equation

k33[t_] :=
20*(-1.01880661301011*
Sin[-12.2868520317448*t]/(12.3729623105373*t +
31.6231386532942*t^2));

int2 = Integrate[x'[tau]*k33[t - tau], {tau, -\[Infinity], t}];

ft = 31.4380978232711 Cos[3.9893240045584673 t];
M33 = 1.95;
m33 = 2.5;

equation = (M33 + m33)*x''[t] + int2 == ft;
ts = 50;

s1 = NDSolve[{equation, x == 1, x' == 1}, x, {t, 0, ts}]
Plot[Evaluate[x[t] /. s1], {t, 0, ts}]
`
• – dcydhb Dec 13 '19 at 11:37
• I'm voting to close this question as off-topic because the problem doesn't lie in Mathematica side, it's because the $B_z$ used by OP doesn't seem to be correct. – xzczd Dec 14 '19 at 3:06
• Comments are not for extended discussion; this conversation has been moved to chat. – Kuba Dec 14 '19 at 12:54