DSolve doesn't work

Question

I need to find a solution for a system of differential equations involving functions $x(t),y(t),z(t)$ with variable $t$. Computing

DSolve[{x''[
t] == (2 y[t] x'[t] y'[t] + 
  x[t] (x'[t]^2 - y'[t]^2 - z'[t]^2))/(x[t]^2 + y[t]^2), 
y''[t] == (2 x[t] x'[t] y'[t] - 
  y[t] (x'[t]^2 - y'[t]^2 + z'[t]^2))/(x[t]^2 + y[t]^2), 
z''[t] == 2 ((x[t] x'[t] + y[t] y'[t]) z'[t])/(x[t]^2 + y[t]^2), 
x'[0] == 1, y'[0] == 1, z'[0] == 1, x[0] == 0, y[0] == 0, 
z[0] == 0}, {x[t], y[t], z[t]}, t]

Mathematica will just repeat my command as output.

Do you have any ideas why Mathematica won't compute? Also, how should I modify my code to get some response?

Thank you!

Answer 1

Edit: Derivation of symbolic solution

As commented by Mariusz Iwaniuk), the initial conditions in the question are inconsistent. Setting {x[0] == 1, y[0] == 1, z[0] == 1} eliminates the inconsistency but does not cause DSolve to return a solution. So, to understand the desired solution, first solve the equations numerically.

s = NDSolveValue[
   {x''[t] == (2 y[t] x'[t] y'[t] + x[t] (x'[t]^2 - y'[t]^2 - z'[t]^2))/(x[t]^2 + y[t]^2), 
    y''[t] == (2 x[t] x'[t] y'[t] - y[t] (x'[t]^2 - y'[t]^2 + z'[t]^2))/(x[t]^2 + y[t]^2), 
    z''[t] == 2 ((x[t] x'[t] + y[t] y'[t]) z'[t])/(x[t]^2 + y[t]^2), 
    x'[0] == 1, y'[0] == 1, z'[0] == 1, x[0] == 1, y[0] == 1, z[0] == 1}, 
   {x, y, z}, {t, -5, 8}];
Plot[Evaluate@Through[s[t]], {t, -5, 8}]

The green curve represents z, and the brown curve represents both x and y, which are identical. Moreover, x and y are identical whenever their initial conditions are identical. This fact allows the equations to be solved symbolically.

Eliminate z (yesterdays result)

The third ODE can be rewritten as

D[z[t], {t, 2}]/D[z[t], t] == D[x[t]^2 + y[t]^2, t]/(x[t]^2 + y[t]^2)

which has as a solution

z'[t] == c (x[t]^2 + y[t]^2)

where c is a constant, which be determined by evaluating the solution at t == 0. For the initial conditions chosen here, c -> 1/2. The first two ODEs can be simplified by means of this first integral.

eq2 ==
{x''[t] == (2 y[t] x'[t] y'[t] + x[t] (x'[t]^2 - y'[t]^2 - z'[t]^2))/(x[t]^2 + y[t]^2), 
 y''[t] == (2 x[t] x'[t] y'[t] - y[t] (x'[t]^2 - y'[t]^2 + z'[t]^2))/(x[t]^2 + y[t]^2)}
      /. z'[t] -> c (x[t]^2 + y[t]^2)
(* {x''[t] == (2 y[t] x'[t] y'[t] + x[t] (x'[t]^2 - y'[t]^2 - c^2 (x[t]^2 + y[t]^2)^2))/
        (x[t]^2 + y[t]^2), 
    y''[t] == (2 x[t] x'[t] y'[t] - y[t] (x'[t]^2 - y'[t]^2 + c^2 (x[t]^2 + y[t]^2)^2))/
        (x[t]^2 + y[t]^2)} *)

The ODE system to be solved has been reduced from sixth to fourth order. Nonetheless, DSolve returns unevaluated from

DSolve[eq2, {x, y}, t]

Obtain solution for x and y, then z

As just noted, insight from the numerical solution suggests the substitution

eq1 = (eq2 /. x -> y) // Simplify//First
(* 2 c^2 y[t]^3 + y''[t] == y'[t]^2/y[t] *)

for which DSolve finally returns an answer.

sol = Simplify[First[y[t] /. 
    DSolve[{eq1}, y[t], t, Assumptions -> y > 0]] /. {C[1] -> c1^2, C[2] -> c2}, c1 > 0]
(* (2 c1^2 E^(c1 (c2 + t)))/(2 c^2 c1^2 + E^(2 c1 (c2 + t))) *)

(The assumption, y > 0, (clearly true from the numerical solution) is essential to obtaining a tractable result.) Next, boundary conditions are imposed.

{sol, D[sol, t]} /. t -> 0;
con = Last[Solve[(% /. c -> 1/2) == 1, {c1, c2}] /. C[1] -> 0]
(* {c1 -> Sqrt[3/2], c2 -> Sqrt[2/3] (-Log[2] + Log[3 - Sqrt[6]])} *)

FullSimplify[sol /. c -> 1/2 /. con]
(* -((6 (-3 + Sqrt[6]) E^(Sqrt[3/2] t))/(3 + (-3 + Sqrt[6])^2 E^(Sqrt[6] t))) *)

which is the desired solution for x and y. z now can be obtained from

DSolveValue[{z'[t] == %, z'[0] == 1, z[0] == 1}, z[t], t] // FullSimplify
(* (-5 - 2 Sqrt[6]) (-5 + 2 Sqrt[6] + 4 (-Sqrt[2] + Sqrt[3]) ArcTan[Sqrt[2] - Sqrt[3]] + 
   2 Sqrt[6 (5 - 2 Sqrt[6])] ArcTan[Sqrt[5 - 2 Sqrt[6]]] + (-10 Sqrt[2] + 
   8 Sqrt[3]) ArcTan[(-Sqrt[2] + Sqrt[3]) E^(Sqrt[3/2] t)]) *)