How to solve the differential equation , when my initial condition is Infinity

Question

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.

Nasser · Answer

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]

This means the solution is
$$
    TD(r)=infty
$$
You probably want to check you BC are correct.

How to solve the differential equation , when my initial condition is Infinity

One Answer

Add your own answers!

Ask a Question