ADDED LATER: READ THE POST SCRIPT TEXT AT THE END
Before saying anything else I would like to say please don't edit my codes in this question. I know that they are not formatted properly but they'll work (or you can say not work) and show the same error message that I get. Just copy and paste. After you paste them in mathematica they'll automatically be formatted as they should be in a nice way. While answering the question you may want to post properly formatted code if you wish.
Okay this is going to be be a complicated question. I have four differential equations with variables $r[s], \theta[s],\phi[s],t[s]$ where $s$ is used to parameterize all equations. The differential equations are humongous but still I am posting it as I cannot figure out where the problem is. Mathematica is considering $\theta$ to be an independent variable but since I have parameterized the equation with $s$ the only independent parameter/variable which should remain is $s$. The error which I am receiving from mathematica while trying to use NDSolve function is
NDSolve::ivhead: The independent variable θ appears in the head of the expression θ[s]. The independent variables should always be arguments.
I am posting the differential equations for if someone can find for me where the error is that'll be great. I am trying to make it work since more than 8 hours and I have lost all patience. All the coordinates are labelled as a function of $s$. The differential equations are correct and the only place the problem is should be labeling in labeling coordinates as $\theta[s]$
$$\text{Radial Equation}$$
radial = ((r[s] (2 a^2 + r[s] (-3 M + r[s])) +
a^2 (M - r[s]) Cos[θ[s]]^2) r'[
s]^2)/((a^2 + r[s] (-2 M + r[s])) (r[s]^2 +
a^2 Cos[θ[s]]^2)) - (
M (a^2 + r[s] (-2 M + r[s])) (-3 r[s]^2 +
a^2 Cos[θ[s]]^2) t'[s]^2)/(r[s]^2 +
a^2 Cos[θ[s]]^2)^5 - (
4 a^2 Cos[θ[s]] Sin[θ[s]] r'[s] θ'[s])/(
r[s]^2 + a^2 Cos[θ[s]]^2) - (
2 r[s] (a^2 + r[s] (-2 M + r[s])) θ'[s]^2)/(
r[s]^2 + a^2 Cos[θ[s]]^2) - (
2 a M (a^2 + r[s] (-2 M + r[s])) (-3 r[s]^2 +
a^2 Cos[θ[s]]^2) Sin[θ[s]]^2 t'[s] ϕ'[
s])/(r[s]^2 +
a^2 Cos[θ[s]]^2)^5 + ((a^2 +
r[s] (-2 M + r[s])) (-2 r[s] Sin[θ[s]]^2 - (
2 a^2 M (-3 r[s]^2 + a^2 Cos[θ[s]]^2) Sin[θ[
s]]^4)/(r[s]^2 + a^2 Cos[θ[s]]^2)^3) ϕ'[s]^2)/(
2 (r[s]^2 + a^2 Cos[θ[s]]^2)^2) + r''[s];
$$\text{Theta Equation}$$
theta = (Sin[2 θ[s]] r'[
s]^2)/((1 + r[s] (-2 M + r[s])) (r[s]^2 +
Cos[θ[s]]^2)) - (
64 M r[s] Cos[θ[s]] Sin[θ[s]] t'[s]^2)/(r[s]^2 +
16 Cos[θ[s]]^2)^5 + (2 r[s] r'[s] θ'[s])/(
r[s]^2 + Cos[θ[s]]^2) + (2 r[s] r'[s] θ'[s])/(
r[s]^2 + 4 Cos[θ[s]]^2) - (
8 Cos[θ[s]] Sin[θ[s]] θ'[s]^2)/(
r[s]^2 + 4 Cos[θ[s]]^2) + (
3 M r[s] Cos[θ[s]] (-27 - 2 r[s]^2 +
9 Cos[2 θ[s]]) Sin[θ[s]] t'[s] ϕ'[
s])/(r[s]^2 + 9 Cos[θ[s]]^2)^5 + (
4 M r[s] Cos[θ[s]] (-48 - 2 r[s]^2 +
16 Cos[2 θ[s]]) Sin[θ[s]] t'[s] ϕ'[
s])/(r[s]^2 + 16 Cos[θ[s]]^2)^5 +
1/(r[s]^2 + 9 Cos[θ[s]]^2)^5 Cos[θ[s]] Sin[θ[
s]] ((9 - r[s]^2) (r[s]^2 + 9 Cos[θ[s]]^2)^3 -
36 M r[s] (r[s]^2 + 9 Cos[θ[s]]^2) Sin[θ[s]]^2 -
324 M r[s] Sin[θ[s]]^4) ϕ'[s]^2 + θ''[s];
$$\text{Phi Equation}$$
phi = (M (-3 r[s]^2 + Cos[θ[s]]^2) r'[s] t'[
s])/((r[s]^2 +
Cos[θ[s]]^2) (r[s] (-1 + r[s]^2) (-2 M + r[s]^3) -
2 r[s]^2 (1 - r[s]^2) Cos[θ[s]]^2 + (-1 +
r[s]^2) Cos[θ[s]]^4 +
2 M r[s] Sin[θ[s]]^2)) + (4 M (-3 r[s]^2 +
16 Cos[θ[s]]^2) r'[s] t'[
s])/((r[s]^2 +
16 Cos[θ[s]]^2) (r[s] (-16 + r[s]^2) (-2 M + r[s]^3) -
32 r[s]^2 (16 - r[s]^2) Cos[θ[s]]^2 + (-4096 +
256 r[s]^2) Cos[θ[s]]^4 +
32 M r[s] Sin[θ[s]]^2)) + (M r[
s] (80 - 16 M r[s] + 64 r[s]^2 + 8 r[s]^4 +
64 Cos[2 θ[s]] - 16 Cos[4 θ[s]]) Cot[θ[
s]] t'[s] θ'[
s])/(2 (r[s]^2 +
4 Cos[θ[s]]^2)^2 (r[s] (-4 + r[s]^2) (-2 M + r[s]^3) -
8 r[s]^2 (4 - r[s]^2) Cos[θ[s]]^2 + (-64 +
16 r[s]^2) Cos[θ[s]]^4 +
8 M r[s] Sin[θ[s]]^2)) + (M r[
s] (1280 - 16 M r[s] + 256 r[s]^2 + 8 r[s]^4 +
1024 Cos[2 θ[s]] -
256 Cos[4 θ[s]]) Cot[θ[s]] t'[s] θ'[
s])/((r[s]^2 +
16 Cos[θ[s]]^2)^2 (r[
s] (-16 + r[s]^2) (-2 M + r[s]^3) -
32 r[s]^2 (16 - r[s]^2) Cos[θ[s]]^2 + (-4096 +
256 r[s]^2) Cos[θ[s]]^4 +
32 M r[s] Sin[θ[s]]^2)) + ((-2 M r[s]^4 + r[s]^7 +
3 r[s]^3 Cos[θ[s]]^4 + r[s] Cos[θ[s]]^6 -
3 M r[s]^2 Sin[θ[s]]^2 +
Cos[θ[s]]^2 (-2 M r[s]^2 + 3 r[s]^5 +
M Sin[θ[s]]^2)) r'[s] ϕ'[
s])/((r[s]^2 +
Cos[θ[s]]^2) (r[s] (-1 + r[s]^2) (-2 M + r[s]^3) -
2 r[s]^2 (1 - r[s]^2) Cos[θ[s]]^2 + (-1 +
r[s]^2) Cos[θ[s]]^4 +
2 M r[s] Sin[θ[s]]^2)) + ((-2 M r[s]^4 + r[s]^7 +
243 r[s]^3 Cos[θ[s]]^4 + 729 r[s] Cos[θ[s]]^6 -
27 M r[s]^2 Sin[θ[s]]^2 +
9 Cos[θ[s]]^2 (-2 M r[s]^2 + 3 r[s]^5 +
9 M Sin[θ[s]]^2)) r'[s] ϕ'[
s])/((r[s]^2 +
9 Cos[θ[s]]^2) (r[s] (-9 + r[s]^2) (-2 M + r[s]^3) -
18 r[s]^2 (9 - r[s]^2) Cos[θ[s]]^2 + (-729 +
81 r[s]^2) Cos[θ[s]]^4 +
18 M r[s] Sin[θ[s]]^2)) - (Csc[θ[
s]] (256 r[s]^2 (4 - r[s]^2) Cos[θ[s]]^7 +
256 (4 - r[s]^2) Cos[θ[s]]^9 +
32 r[s] Cos[θ[s]]^5 (r[s]^2 (M - 3 r[s]^3) +
4 (-2 M + 3 r[s]^3) + 4 M Cos[2 θ[s]]) -
4 Cos[θ[s]]^3 (4 r[s]^3 (4 - r[s]^2) (M - r[s]^3) +
64 M r[s] Sin[θ[s]]^4) -
r[s]^2 Cos[θ[
s]] (r[s]^3 (-4 + r[s]^2) (-2 M + r[s]^3) -
16 M (M - r[s]^3) Sin[θ[s]]^2 +
64 M r[s] Sin[θ[s]]^4 +
32 M r[s] Sin[2 θ[s]]^2)) θ'[s] ϕ'[
s])/((r[s]^2 +
4 Cos[θ[s]]^2)^2 (r[s] (-4 + r[s]^2) (-2 M + r[s]^3) -
8 r[s]^2 (4 - r[s]^2) Cos[θ[s]]^2 + (-64 +
16 r[s]^2) Cos[θ[s]]^4 +
8 M r[s] Sin[θ[s]]^2)) - (Csc[θ[
s]] (2916 r[s]^2 (9 - r[s]^2) Cos[θ[s]]^7 +
6561 (9 - r[s]^2) Cos[θ[s]]^9 +
162 r[s] Cos[θ[s]]^5 (r[s]^2 (M - 3 r[s]^3) +
9 (-2 M + 3 r[s]^3) + 9 M Cos[2 θ[s]]) -
4 Cos[θ[s]]^3 (9 r[s]^3 (9 - r[s]^2) (M - r[s]^3) +
729 M r[s] Sin[θ[s]]^4) -
r[s]^2 Cos[θ[
s]] (r[s]^3 (-9 + r[s]^2) (-2 M + r[s]^3) -
36 M (M - r[s]^3) Sin[θ[s]]^2 +
324 M r[s] Sin[θ[s]]^4 +
162 M r[s] Sin[2 θ[s]]^2)) θ'[s] ϕ'[
s])/((r[s]^2 +
9 Cos[θ[s]]^2)^2 (r[s] (-9 + r[s]^2) (-2 M + r[s]^3) -
18 r[s]^2 (9 - r[s]^2) Cos[θ[s]]^2 + (-729 +
81 r[s]^2) Cos[θ[s]]^4 +
18 M r[s] Sin[θ[s]]^2)) + ϕ''[s];
$$\text{Time Equation}$$
time = (M (1 - r[s]^2) (-3 r[s]^2 + Cos[θ[s]]^2) r'[s] t'[
s])/((r[s]^2 +
Cos[θ[s]]^2) (r[s] (-1 + r[s]^2) (-2 M + r[s]^3) -
2 r[s]^2 (1 - r[s]^2) Cos[θ[s]]^2 + (-1 +
r[s]^2) Cos[θ[s]]^4 +
2 M r[s] Sin[θ[s]]^2)) + (M (16 - r[s]^2) (-3 r[s]^2 +
16 Cos[θ[s]]^2) r'[s] t'[
s])/((r[s]^2 +
16 Cos[θ[s]]^2) (r[s] (-16 + r[s]^2) (-2 M + r[s]^3) -
32 r[s]^2 (16 - r[s]^2) Cos[θ[s]]^2 + (-4096 +
256 r[s]^2) Cos[θ[s]]^4 +
32 M r[s] Sin[θ[s]]^2)) + (4 M r[
s] (16 + 2 M r[s] + 4 r[s]^2 - 2 r[s]^4 +
4 (4 - r[s]^2) Cos[2 θ[s]]) Sin[2 θ[s]] t'[
s] θ'[
s])/((r[s]^2 +
4 Cos[θ[s]]^2)^2 (r[s] (-4 + r[s]^2) (-2 M + r[s]^3) -
8 r[s]^2 (4 - r[s]^2) Cos[θ[s]]^2 + (-64 +
16 r[s]^2) Cos[θ[s]]^4 +
8 M r[s] Sin[θ[s]]^2)) + (16 M r[
s] (256 + 2 M r[s] + 16 r[s]^2 - 2 r[s]^4 +
16 (16 - r[s]^2) Cos[2 θ[s]]) Sin[2 θ[s]] t'[
s] θ'[
s])/((r[s]^2 +
16 Cos[θ[s]]^2)^2 (r[
s] (-16 + r[s]^2) (-2 M + r[s]^3) -
32 r[s]^2 (16 - r[s]^2) Cos[θ[s]]^2 + (-4096 +
256 r[s]^2) Cos[θ[s]]^4 +
32 M r[s] Sin[θ[s]]^2)) + (M (1 - 5 r[s]^2 +
10 r[s]^4 + (1 + r[s]^2) Cos[2 θ[s]]) Sin[θ[
s]]^2 r'[s] ϕ'[
s])/(2 (r[s]^2 +
Cos[θ[s]]^2) (r[s] (-1 + r[s]^2) (-2 M + r[s]^3) -
2 r[s]^2 (1 - r[s]^2) Cos[θ[s]]^2 + (-1 +
r[s]^2) Cos[θ[s]]^4 +
2 M r[s] Sin[θ[s]]^2)) + (3 M (81 - 45 r[s]^2 +
10 r[s]^4 + 9 (9 + r[s]^2) Cos[2 θ[s]]) Sin[θ[
s]]^2 r'[s] ϕ'[
s])/(2 (r[s]^2 +
9 Cos[θ[s]]^2) (r[s] (-9 + r[s]^2) (-2 M + r[s]^3) -
18 r[s]^2 (9 - r[s]^2) Cos[θ[s]]^2 + (-729 +
81 r[s]^2) Cos[θ[s]]^4 +
18 M r[s] Sin[θ[s]]^2)) + (16 M r[
s] Cos[θ[s]] (16 + 2 M r[s] + 4 r[s]^2 - 2 r[s]^4 +
4 (4 - r[s]^2) Cos[2 θ[s]]) Sin[θ[
s]]^3 θ'[s] ϕ'[
s])/((r[s]^2 +
4 Cos[θ[s]]^2)^2 (r[s] (-4 + r[s]^2) (-2 M + r[s]^3) -
8 r[s]^2 (4 - r[s]^2) Cos[θ[s]]^2 + (-64 +
16 r[s]^2) Cos[θ[s]]^4 +
8 M r[s] Sin[θ[s]]^2)) + (54 M r[
s] Cos[θ[s]] (81 + 2 M r[s] + 9 r[s]^2 - 2 r[s]^4 +
9 (9 - r[s]^2) Cos[2 θ[s]]) Sin[θ[
s]]^3 θ'[s] ϕ'[
s])/((r[s]^2 +
9 Cos[θ[s]]^2)^2 (r[s] (-9 + r[s]^2) (-2 M + r[s]^3) -
18 r[s]^2 (9 - r[s]^2) Cos[θ[s]]^2 + (-729 +
81 r[s]^2) Cos[θ[s]]^4 +
18 M r[s] Sin[θ[s]]^2)) + t''[s];
These are the $4$ 2nd order ordinary differential equations which I am trying to solve simultaneously using NDSolve. The code for that is
NDSolve[{time == 0, radial == 0, phi == 0, theta == 0,
r[0] == 8, ϕ[0] == 0, t[0] == 0, ϕ'[0] == 0.1, t'[0] == 0,
r'[0] == 0.1}, r, t, ϕ, θ, {s, 0, 1}]
Post Script: I tried to do the same with equations of similar form and order and got the same error. Since I am getting the same error which means equations does not matter I am doing something fundamentally wrong. I am posting those equation so that you don't have to deal with complicated equations above
aa = x[s] Cos[y'[s] z[s]] + t''[s];
bb = Sin[x'[s]] Cos[y[s]] + t[s] + z''[s];
cc = x[s]^2 + Sin[z'[s]] Exp[t[s]] + y''[s];
dd = z'[s] + x'[s] y[s] + x''[s];
NDSolve[{aa == 0, bb == 0, cc == 0, dd == 0, x[0] == 1, y[0] == 1,
z[0] == 0, t[0] == 0, x'[0] == 1, y'[0] == 1, z'[0] == 0,
t'[0] == 0}, x, y, z, t, {s, 0, 1}]
NDSolve[{aa == 0, bb == 0, cc == 0, dd == 0, x[0] == 1, y[0] == 1, z[0] == 0, t[0] == 0, x'[0] == 1, y'[0] == 1, z'[0] == 0, t'[0] == 0}, {x, y, z, t}, {s, 0, 1}]? $\endgroup$