# Find trajectory of particle solving ODE containg integral

Please suggest how to solve below linear ODE (I want to plot the r vs t graph), I am trying using Integrate:

    f[r_]: = A0/r^3; A0=10^-10;μ = 0.001; b = 4*Pi*a*μ;a=2*10^-6;
f[r] + b*r'[t] == 0
Integrate[1/f[r], r] + Integrate[1/b, t] == c0

data = {{7.*10^-6, -0.317175}, {7.7*10^-6, -2.59621}, {8.4*10^-6,
\
-4.03621}, {9.1*10^-6, -4.79394}, {9.8*10^-6, -5.0422},
{0.0000105, \
-4.94287}, {0.0000112, -4.63177}, {0.0000119, -4.21503},
{0.0000126, \
-3.76242}, {0.0000133, -3.32148}, {0.000014, -2.91612},
{0.0000147,
\
-2.55617}, {0.0000154, -2.24334}, {0.0000161, -1.97411},
{0.0000168, \
-1.74391}, {0.0000175, -1.5469}, {0.0000182, -1.37765},
{0.0000189, \
-1.232}, {0.0000196, -1.10604}, {0.0000203, -0.996608},
{0.000021, \
-0.901123}, {0.0000217, -0.817417}, {0.0000224, -0.743742}, \
{0.0000231, -0.678649}, {0.0000238, -0.620934}, {0.0000245, \
-0.569525}, {0.0000252, -0.523687}, {0.0000259, -0.48258}, \
{0.0000266, -0.445732}, {0.0000273, -0.412469}, {0.000028,
-0.382433}};

f0 = Interpolation[N[data], InterpolationOrder -> 3];
f[r_] = f0[r]


• Rp is not defined ?.Your function is: z[r] or: z[t] ? Decide ? Feb 10, 2023 at 17:08
• Function definitions are written: z[r_] ... Feb 10, 2023 at 19:50
• @Iwaniuk, I have updated the question and corrected the typo Rp. Feb 11, 2023 at 4:40
• What is fc[r]? IS it supposed to be c f[r]?
– bmf
Feb 19, 2023 at 3:51
• @bmf it was typo. Updated it. Feb 19, 2023 at 3:59

I don't really understand what the issue is here. Mathematica is happy to DSolve it, unless I am missing something

f[r_] := A0/r[t]^3;
A0 = 10^-10;
μ = 0.001;
b = 4*Pi*a*μ;
a = 2*10^-6;
ode = f[r] + b*r'[t] == 0;

DSolve[ode, r[t], t] // Rationalize[#, 0] &


• Ok, If I have interpolated data for f(r) then how we can get the solution? updated the question. Feb 19, 2023 at 4:05
• @GopalVerma Then you do NDSolve for the numerical solution.
– bmf
Feb 19, 2023 at 4:06
• I tried but was not able to get the solution. Feb 19, 2023 at 4:10