I can't get DSolve to work on coupled equations

Question

In trying to solve a geodesic equation, I have these coupled equations:

eq1 = Derivative[2][u][l] - Cos[u[l]]*Sin[u[l]]*D[v[l], l]*D[v[l], l] == 0; 
eq2 = Derivative[2][v][l] + 2*Cot[u[l]]*D[u[l], l]*D[v[l], l] == 0; 
eq3 = u[0] == 1; 
eq4 = v[0] == 1; 
eq5 = Derivative[1][u][0] == 1; 
eq6 = Derivative[1][v][0] == 1; 

Now I want a solution for the functions u, v. So I try this:

DSolve[{eq1, eq2, eq3, eq4, eq5, eq6}, {u, v}, l]

And I get:

DSolve[{-Cos[u[l]] Sin[u[l]] Derivative[1][v][l]^2 + (u^\[Prime]\[Prime])[l] == 0, 2 Cot[u[l]] Derivative[1][u][l] Derivative[1][v][l] + (v^\[Prime]\[Prime])[l] == 0, u[0] == 1, v[0] == 1, Derivative[1][u][0] == 1, Derivative[1][v][0] == 1}, {u, v}, l]

Which just tells me that Mathematica didn't recognize my input. What am I doing wrong?

Answer 1

DSolve often cannot solve differential equations for which symbolic solutions actually do exist. So, out of curiosity, I attempted to solve these equations and made some progress, obtaining v'[l] in terms of u[l], and u[l] as an implicit function of l. Here is the calculation.

Obtain v'[l]:

Equal @@ (DSolve[eq2, u[l], l][[1, 1]]) /. Exp[2 C[1]] -> c
Solve[%, v'[l]][[1, 1]] /. Exp[2 C[1]] -> c
% /. l -> 0 /. (Rule @@@ {eq3, eq6})
Reverse@Simplify[Sin[1]^2 # & /@ %]
svp = %%% /. %
(* v'l] -> Csc[u[l]]^2 Sin[1]^2 *)

Next obtain a first integral for u[l].

Equal @@ (Solve[eq1 /. svp, u''[l]][[1, 1]])
(* (u''[l] == Cot[u[l]] Csc[u[l]]^2 Sin[1]^4 *)
u'[l]^2 == 2 Integrate[%[[2]], u[l]] + c
Solve[FullSimplify[% /. l -> 0 /. (Rule @@@ {eq3, eq5})], c][[1]]
Equal @@ (Solve[Simplify[%% /. %], u'[l]][[2, 1]])
(* u'[l] == Sqrt[1 + Sin[1]^2 - Csc[u[l]]^2 Sin[1]^4] *)

There is, of course, a second solution with the right side of the equation negative. Finally, solve the last equation to obtain

(# - c) & /@ FullSimplify[Integrate[1/%[[2]], u[l]] == l + c, u {l} > 0]
Solve[FullSimplify[% /. l -> 0 /. (Rule @@ eq3)], c][[1]]
sp = %% /. %
(* l == ArcTan[Sqrt[1/2 (3 - Cos[2])] Cot[1]] Sqrt[(1 - Cos[2])/(3 - Cos[2])] Csc[1] 
   - (4 ArcTan[(2 Cos[u[l]])/Sqrt[4 + 8/(-3 + Cos[2]) + 2 Cos[2] - 2 Cos[2 u[l]]]] 
   Sqrt[2 + 4/(-3 + Cos[2]) + Cos[2] - Cos[2 u[l]]] 
   Sqrt[3 - Cos[2] - 2 Csc[u[l]]^2 Sin[1]^4] Sin[u[l]])/
   (3 + 2 Cos[2] - Cos[4] + 2 (-3 + Cos[2]) Cos[2 u[l]]) *)

which is the desired solution for u[l] with u'[l] > 0, though implicit. The solution for u'[l] < 0 is obtained similarly.

For comparision, the numerical solution for u[l] and v'[l] is

NDSolveValue[{eq1, eq2, eq3, eq4, eq5, eq6}, {u[l], v'[l]}, {l, -2, 8}];
Plot[{Evaluate@%, Sin[1]^2 Csc[First[%]]^2}, {l, -2, 8}, ImageSize -> Large, 
    AxesLabel -> {"u,v'", l}, LabelStyle -> {15, Bold, Black}]

The analytical solutions compare well with this. In summary, it is possible to obtain symbolic solutions for these equations with Mathematica, although the solution for u[l] is implicit. All things considered, it seems understandable that DSolve could not provide a viable solution for the equations, although showing relevant intermediate results would have been helpful.