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Problem of `Integrate` and `DSolve` with inhomogeneous Legendre differential equation


I have the following inhomogeneous Legendre differential equation $(m=n=2)$:


 deq=(6 - 4/(1 - z^2)) h2[z] - 
  2 z Derivative[1][h2][z] + (1 - z^2) Derivative[2][h2][z] == (
  6 (-1 + z + z^2) μ^2)/(M^4 (-1 + z^2)^2) + (

  3 μ^2 Log[(-1 + z)/(1 + z)])/(M^4 (-1 + z))

where the variable $z>1$. I can not get Mathematica to give me the simplest/correct solutions. I think it has to do with those logarithms. I know that the homogeneous solution is


solH=A*LegendreQ[2,2,3,z]+B*LegendreP[2,2,3,z]

I need mathematica's type 3 Legendre functions because for $z>1$ the type 3 functions are on the right branch. With DSolve however I can only get the expanded forms of type 1:


DSolve[deq[[1]] == 0, h2[z], z]

gives back


 {h2[z] -> -3 (-1 + z^2) C[1] + C[2] (((1 - z^2) (5 z - 3 z^3))/(-1 + z^2)^2 + 

      3 (1 - z^2) (-(1/2) Log[1 - z] + 1/2 Log[1 + z]))}

which is not real for $z>1$. Is there a way to tell DSolve to work with $z>1$ or to search for real solutions under that assumption? $Assumptions={z>1} alone does not do the trick.


Apart from that I have a bigger problem with the particular solution: DSolve[deq, h2[z], z][[1]] gives a ridiculous partial solution which is incredibly long and complex. I want a real particular solution; I want a solution that is real for all $z>1$. I know from literature that there is one (plugging that one in proves that it is indeed one):


{h2p[z] -> -3 μ^2/(16 M^4) (6 z^2 + 3 z - 6 - (4 z^2 + 2 z)/(z^2 - 1)) - 3 μ^2/(32 M^4) (3 z^2 - 8 z - 3 - 8/(z^2 - 1)) Log[(z - 1)/(z + 1)] + 3 μ^2/(16 M^4) (z^2 - 1) Log[(z - 1)/(z + 1)]^2}

Again is there a way to tell DSolve to produce a real solution for $z>1$? I also tried constructing that particular solution using the Variation of Parameters method


WLQ223LP223 = 
 Wronskian[{LegendreQ[2, 2, 3, z], LegendreP[2, 2, 3, z]}, z]
-LegendreQ[2, 2, 3, z]*

    Integrate[deq[[2]]*LegendreP[2, 2, 3, z]/WLQ223LP223, z] + 
   LegendreP[2, 2, 3, z]*
    Integrate[deq[[2]]*LegendreQ[2, 2, 3, z]/WLQ223LP223, z] // 
  FunctionExpand // Simplify

Mathematica finds an analytical solutions but they do not reduce to the particular solution I want: the involved integrals become very lengthy and again complex.


Anyone around here who has some experience with such differential equations in Mathematica?


I send the problem to a friend who has Maple and Maple's dsolve finds the "correct"/simple partial solution, when one prepends assume(z > 1). It does so using the Variation of Parameters method and the solutions of the homogeneous equation.



Answer



Personally, I prefer @corey979's approach (+1), but with the following tweaks.



solExpr = 
  Assuming[z > 1 && M ∈ Reals && μ ∈ Reals,
   DSolveValue[deq, h2[z], z] /. {Log[-1 + z^2] -> Log[-1 + z] + Log[1 + z], 
        Log[(-1 + z)/(1 + z)] -> Log[-1 + z] - Log[1 + z]} // Re // 
     ComplexExpand[#, TargetFunctions -> {Re, Im}] & // Simplify
   ];

sol = h2 -> Function @@ {z, solExpr}

Mathematica graphics



deq /. sol // Simplify[#, z > 1 && M ∈ Reals && μ ∈ Reals] &
(*  True  *)

Plot[h2[z] /. sol /. {M -> 1, μ -> 1, C[1] -> -1, C[2] -> 1},
 {z, 1, 5}]

Mathematica graphics


The main tweaks are including all assumptions to make all parameters real in both DSolve and Simplify and the use of TargetFunctions. It's also easier to simplify an algebraic expression; hence DSolveValue[]. The rules


{Log[-1 + z^2] -> Log[-1 + z] + Log[1 + z], 
 Log[(-1 + z)/(1 + z)] -> Log[-1 + z] - Log[1 + z]}


help with simplification; they are not important and may be omitted.


The main reason I prefer this is that the real and imaginary parts are independently solutions of a real ODE, a key point made by corey979. It just seems a simpler approach, when it simplifies down nicely. In this case it does, and we get a general solution that spans the solution space.


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