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int[r_]=-((557.310989080004 r)/((37.3042 - r) (-25.578 + r) (62.8822 + 
r) Sqrt[0.0345106153943703 - ((37.3042 - r) (-25.578 + 
  r) (62.8822 + r))/(3000 r)]))

This is the function of r , now next is that I want to solve the differential equation using this function. Before that at my boundary conditions, this whole function vanishes as;

int[maxr]=1/0
int[rg]=1/0

maxr=31.0723250284387 and rg=25.57799942210616`

Next I try to solve the following differential equation via NDsolve

sol = NDSolve[{TD'[r] == int[r], TD[maxr] == Infinity}, {TD}, {r,maxr, rg}, Method->"ExplicitRungeKutta"]

But it fails are throws some errors like Solve::infc: The system TD==[Infinity] contains an infinite object [Infinity]. NDSolve::icfail: Unable to find initial conditions that satisfy the residual function within specified tolerances. Try giving initial conditions for both values and derivatives of the functions. Can anyone help me out.

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The solution to the ODE is infinity.

You can see this by solving analytically

ClearAll[r, TD, k];
intr = -((557.310989080004 r)/((37.3042 - r) (-25.578 + r) (62.8822 + 
         r) Sqrt[0.0345106153943703 - ((37.3042 - r) (-25.578 + 
              r) (62.8822 + r))/(3000 r)]));
intr = SetPrecision[intr, Infinity];
maxr = SetPrecision[31.0723250284387, Infinity];
rg = SetPrecision[25.57799942210616, Infinity];
sol = TD[r] /. First@DSolve[{TD'[r] == intr}, TD[r], r];

Now set up an equation to solve for constant of integration

eq = (sol /. r -> maxr) == k;
constant = C[1] /. First@N@Solve[eq, C[1]];

Replace the constant found back in the solution

sol = sol /. C[1] -> constant;

Now let k go to infinity, which maps to BC condition that TD[maxr] == Infinity

Limit[sol, k -> Infinity]

Mathematica graphics

This means the solution is

$$ TD(r)=\infty $$

You probably want to check you BC are correct.

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  • $\begingroup$ Thanks for clarfication. But finally i want to plot TD[r], from maxr to rg . Which should be diverging at maxr and rg. How can that be done $\endgroup$
    – Immy Salam
    Apr 26 at 13:10
  • $\begingroup$ @ImmySalam Plot the result of NDSolve[{TD'[r] == int[r], TD[28] == 0}, TD[r], {r, 25, 32}], which goes to Infinity at rg and -Infinity at rmax. Changing the value of TD[28] simply moves the whole curve up or down but does not change its shape. $\endgroup$
    – bbgodfrey
    Apr 26 at 13:24

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