# How to solve this differential equation analytically?

How can I DSolve the following differential equation? Is there an analytical answer to this equation?

DSolve[(1/(G*L*(n - r)*(n + r)))*Sin[\[Theta]1]*Sin[\[Theta]2]*
(54*(n^2 - r^2)^6 + 2*L^2*(9*r^4*(9*q0^2 + 2*r^6) +
36*n^6*r^4*(-5 + 4*V0^2) -
18*n^10*(1 + 4*V0^2) + 18*n^8*r^2*
(5 + 16*V0^2) - 2*n^2*(9*q0^2*r^2 +
96*q0*r^5*V0 + r^8*(45 + 4*V0^2)) +
n^4*(9*q0^2 - 192*q0*r^3*V0 +
20*r^6*(9 + 8*V0^2))) +
9*L^2*(12*L^4*m*(14*n^6 - 51*n^4*r^2 + 5*r^6)*
V[r]^3 + (n - r)^3*(n + r)^3*Derivative[1][V][
r]*(4*r*(n^2 - r^2)^2 + 3*L^4*m*
Derivative[1][V][r]*(n^2 + r^2 +
(3*n^2*r + r^3)*Derivative[1][V][r])) +
6*L^4*m*(n - r)*(n + r)*V[r]^2*
(12*n^2*r*(-5*n^2 + r^2)*Derivative[1][V][r] +
(n - r)*(n + r)*(10*n^2 - 6*r^2 -
6*(n^4 - n^2*r^2 + 2*r^4)*Derivative[2][V][
r] + 3*(n - r)*r*(n + r)*(2*n^2 + r^2)*
Derivative[3][V][r])) - (n - r)^2*(n + r)^2*
V[r]*(4*(n^6 + (-3*L^4*m + n^4)*r^2 -
5*n^2*r^4 + 3*r^6) + 3*L^4*m*
(-36*n^2*r^2*Derivative[1][V][r]^2 +
(n - r)*(n + r)*(2*(n^2 + 4*r^2)*
Derivative[2][V][r] + 3*r^2*(-n + r)*
(n + r)*Derivative[2][V][r]^2 +
2*r*(-n + r)*(n + r)*Derivative[3][V][
r]) + 3*r*Derivative[1][V][r]*
(8*n^2 - 2*(4*n^4 - 5*n^2*r^2 + r^4)*
Derivative[2][V][r] - r*(n^2 - r^2)^2*
Derivative[3][V][r]))))) == 0,
V[r], r]

• Your code dosen't work for me. Check syntax.When Mathematica returns the input as the output, it means that the calculation returned unevaluated. This often means that the function does not have the methods available to solve the problem symbolically, or it is mathematically impossible to obtain a symbolic solution (not all sums, integrals, or differential equations have symbolic solutions after all). – Mariusz Iwaniuk Jul 14 at 9:52
• What about starting with a simpler equation and a correct MA syntax? Your present code does not represent a valid MA expression. – yarchik Jul 14 at 9:54
• For me, it is ok in any version! I will attach an image too. – Perfect Fluid Jul 14 at 10:34
• With your updated's code give me an error DSolve::dvnoarg: The function V appears with no arguments. ?.Maybe you can try: DSolve[eq // InputForm, V[r], r] and then copy/paste ? – Mariusz Iwaniuk Jul 14 at 10:53
• @PerfectFluid Any reason to expect an analytic closed-form solution exists? Equations do not, in general, have "simple" solutions. Numeric methods are all we can do, usually. – AccidentalFourierTransform Jul 14 at 14:54

Just to get you started: it is not the full solution, though... Let us denote your equation as eq. Then for a partial case of $$m=0$$, there is a nice analytic solution
DSolve[Simplify[eq/.{m->0}],V[r],r]

• Tank you, but actually this is not the case. The most important part of this problem is m and it should be survived. – Perfect Fluid Jul 17 at 9:04
• @PerfectFluid If other parameters are sufficiently small it might be worth it to try AsymptoticDSolveValue, otherwise you are probably better off with a numeric solution. – Thies Heidecke Jul 17 at 15:33