For my research project I need to solve the following integral
I =
1/2 Integrate[π Sqrt[r[θ]^2 Sin[θ]^2]
Sqrt[r[θ]^2 + Derivative[1][r][θ]^2/
(1 + 1/r[θ]^2 - 3/r[θ] + r[θ]^2)],
{θ, 0, 0.005}]
where r[θ]
is a solution of following differential equation
-2 r[θ] + 12 r[θ]^2 - 22 r[θ]^3 + 12 r[θ]^4 - 6 r[θ]^5 + 12 r[θ]^6 -
4 r[θ]^7 - 2 r[θ]^9 +
(Cot[θ] r[θ]^2 - 3 Cot[θ] r[θ]^3 + Cot[θ] r[θ]^4 + Cot[θ] r[θ]^6)
Derivative[1][r][θ] +
(-2 r[θ] + (15 r[θ]^2)/2 - 3 r[θ]^3 - 4 r[θ]^5) Derivative[1][r][θ]^2 +
Cot[θ] r[θ]^2 Derivative[1][r][θ]^3 +
(r[θ]^2 - 3 r[θ]^3 + r[θ]^4 + r[θ]^6) Derivative[2][r][θ]
with initial conditions {r[0] == r0, r'[0] == 0}
.
It's a second order non-linear differential equation. The independent variable θ runs from θ = {0, 0.005}
. There is a singularity at θ = 0
, which can be avoided by taking θ = 10^-10
(some small value). Then initial conditions modify to {r[10^-10] == r0, r'[10^-10] == 0}
.
I have to choose r0
such that by solving the above equation one should get r[0.005] = 10000
. The initial guess for r0
(depending upon the working precision) can be around r0 = 199.958344
.
At the end, integral should gives value I = 76.96884
.
I am solving this problem in Mathematica and getting answer like I = 76.9844
.
Can you please help me in solving this problem.
==
somewhere. $\endgroup$(r^′′)[θ]
in the above, do you really meanDerivative[2][r][θ]
? $\endgroup$