Solve differential equation with constant

Question

How can I get a symbolic solution from the following differential equation? I don't know in that case using a change of variable is a good idea. I have tried in maple as well but the form of the solution was not in my desire.

wave = (3 Derivative[1][a][t]/a[t] Derivative[1][T][t]) + 
   T[t] (\[CapitalOmega]^2/
      a[t]^2 - (6 Derivative[1][a][t]^2/
        a[t]^2) - (2 (a^\[Prime]\[Prime])[t]/a[t])) + (
    T^\[Prime]\[Prime])[t] == 0
a[t_] := t^\[Alpha];
DSolve[(-((2 (-1 + \[Alpha]) \[Alpha])/t^2) - (6 \[Alpha]^2)/t^2 + 
      t^(-2 \[Alpha]) \[CapitalOmega]^2) T[t] + (
   3 \[Alpha] Derivative[1][T][t])/t + (T^\[Prime]\[Prime])[t] == 
  0, {T[t], T[t]}, {t}]

Thank you!

Answer 1

DSolve can solve if we perform some algebraic manipulations:

eq = (-((2 (-1 + \[Alpha]) \[Alpha])/t^2) - (6 \[Alpha]^2)/t^2 + 
 t^(-2 \[Alpha]) \[CapitalOmega]^2) T[t] + (3 \[Alpha] T'[t])/t + T''[t];
eq1 = (eq*t^2 // Simplify // Expand) == 0

$$\Omega ^2 t^{2-2 \alpha } T(t)+t^2 T''(t)+3 \alpha t T'(t)-8 \alpha ^2 T(t)+2 \alpha T(t)=0$$

If we change exponent from: $2-2 \alpha$ to $z$ then:

 eq3 = 2 \[Alpha] T[t] - 8 \[Alpha]^2 T[t] + 
t^z \[CapitalOmega]^2 T[t] + 3 t \[Alpha] T'[t] + t^2 T''[t] == 0;
DSolve[eq3, T[t], t] /. z -> 2 - 2 \[Alpha] // FullSimplify
(*Go back from z to 2-2 \alpha*)

$$T(t)=\left(t^{2-2 \alpha }\right)^{\frac{3}{4}+\frac{1}{2 (-1+\alpha )}} (2-2 \alpha )^{-\frac{3}{2}+\frac{1}{1-\alpha }} \Omega ^{\frac{3}{2}+\frac{1}{-1+\alpha }} \left(J_{\frac{\sqrt{1+\alpha (-14+41 \alpha )}}{2-2 \alpha }}\left(\frac{\sqrt{t^{2-2 \alpha }} \Omega }{1-\alpha }\right) c_2 \Gamma \left(1+\frac{\sqrt{1+\alpha (-14+41 \alpha )}}{2-2 \alpha }\right)+J_{\frac{\sqrt{1+\alpha (-14+41 \alpha )}}{2 (-1+\alpha )}}\left(\frac{\sqrt{t^{2-2 \alpha }} \Omega }{1-\alpha }\right) c_1 \Gamma \left(1+\frac{\sqrt{1+\alpha (-14+41 \alpha )}}{2 (-1+\alpha )}\right)\right)$$

Answer 2

Mathematica can't solve this

ClearAll[α, t, Ω, T]
ode = (-((2 (-1 + α) α)/t^2) - (6 α^2)/t^2 + t^(-2 α) Ω^2) T[t] + (3 α T'[t])/t + T''[t] == 0;
DSolve[ode, T[t], t]

It can however solve it when α is given a specific value

 DSolve[ode /. {α -> 1}, T[t], t]

 DSolve[ode /. {α -> 2}, T[t], t]

DSolve[ode /. {α -> 1/2}, T[t], t]

You said you tried Maple but did not like its solution. Why? Maple can solve this without given specific value for α and its solution agrees with Mathematica above for specific values of α :

local Omega:
ode := (-((2 (-1 + alpha)*alpha)/t^2) - (6*alpha^2)/t^2 + t^(-2* alpha)* Omega^2)*T(t)+ (3 *alpha* diff(T(t),t))/t + diff(T(t),t$2)= 0;
dsolve(ode)

This seems to be limitation of DSolve at this time.