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I am trying to solve this differential equation but Mathematica just return the same code...

enter image description here

Here is the code:

DSolve[{F5'[x] == F4[x], F1'[x] == a x F4[x], 
F4'[x] + F1[x] + a x F5[x] == 0}, {F1[x], F4[x], F5[x]}, x]

I don't know what's wrong... Any ideas? Thank you!!!

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    $\begingroup$ Your code is correct. DSolve[]'s support for System of differential equations is still somewhat limited, so don't be surprised if some things don't work yet.If a symbolic solver returns unevaluated, then Mathematica can't solve the problem.I tried with Maple and Can solve.Maybe this will interest you: 12000.org/my_notes/Murphy_ODE/KEse1.htm#x3-20001 $\endgroup$ May 9 '21 at 9:27
  • $\begingroup$ Ok, thank you! If Maple can solve it, can you copy here the solution? Thanks! $\endgroup$
    – Gabri
    May 9 '21 at 9:46
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With little help and algebraic manipulation we can solve:

 eq1 = F5'[x] - F4[x] == 0;
 eq2 = F1'[x] - a*x*F4[x] == 0;
 eq3 = F4'[x] + F1[x] + a*x*F5[x] == 0;
 EQ = D[eq3, x] /. F1'[x] -> a*x*F4[x] /. F4''[x] -> F5'''[x] /. F4[x] -> F5'[x]
 f5 = F5[x] /. First@DSolve[EQ, F5[x], x][[1]] // Simplify(*Solution for F5[x]*)
 f4 = D[f5, x] // Simplify(*Solution for F4[x]*)
 f1 = (F1[x] /. First@Solve[eq3, F1[x]]) /. (F5[x] -> f5) /. F4'[x] -> D[f4, x] // Simplify(*Solution for F1[x]*)
 

And the end we can check if solutions are TRUE

{eq1 /. F5'[x] -> D[f5, x] /. F4[x] -> f4, 
eq2 /. F1'[x] -> D[f1, x] /. F4[x] -> f4, 
eq3 /. F4'[x] -> D[f4, x] /. F1[x] -> f1 /. F5[x] -> f5} // Simplify

{True, True, True}

UPDATE: 16.05.2021

Mathematica 12.3 can solve. I tried on Wolfram Cloud.

DSolve[{F5'[x] == F4[x], F1'[x] == a x F4[x], F4'[x] + F1[x] + a x F5[x] == 0}, {F1[x], F4[x], F5[x]}, x] 

  {{F1[x] -> C[1]*HypergeometricPFQ[{-1/6}, {-1/3, 1/3}, (-2*a*x^3)/9] + 
 (2^(2/3)*a^(2/3)*x^2*C[2]*HypergeometricPFQ[{1/2}, {1/3, 5/3}, 
 (-2*a*x^3)/9])/(3*3^(1/3)) + 
 (2*2^(1/3)*a^(4/3)*x^4*C[3]*HypergeometricPFQ[{7/6}, {5/3, 7/3}, 
 (-2*a*x^3)/9])/(9*3^(2/3)), 
 F4[x] -> -(x*C[1]*HypergeometricPFQ[{5/6}, {2/3, 4/3}, (-2*a*x^3)/9]) + 
 (C[2]*((2*2^(2/3)*a^(2/3)*x*HypergeometricPFQ[{1/2}, {1/3, 5/3}, 
 (-2*a*x^3)/9])/(3*3^(1/3)) - 
   (2^(2/3)*a^(5/3)*x^4*HypergeometricPFQ[{3/2}, {4/3, 8/3}, 
 (-2*a*x^3)/9])/(5*3^(1/3))))/(a*x) + 
 (C[3]*((8*2^(1/3)*a^(4/3)*x^3*HypergeometricPFQ[{7/6}, {5/3, 7/3}, 
 (-2*a*x^3)/9])/(9*3^(2/3)) - 
   (2*2^(1/3)*a^(7/3)*x^6*HypergeometricPFQ[{13/6}, {8/3, 10/3}, 
 (-2*a*x^3)/9])/(45*3^(2/3))))/
 (a*x), F5[x] -> C[1]*(-(HypergeometricPFQ[{-1/6}, {-1/3, 1/3}, 
 (-2*a*x^3)/9]/(a*x)) + 
  HypergeometricPFQ[{5/6}, {2/3, 4/3}, (-2*a*x^3)/9]/(a*x) - 
  (5*x^2*HypergeometricPFQ[{11/6}, {5/3, 7/3}, (-2*a*x^3)/9])/8) + 
 C[2]*(-1/3*(2^(2/3)*x*HypergeometricPFQ[{1/2}, {1/3, 5/3}, 
 (-2*a*x^3)/9])/(3^(1/3)*a^(1/3)) + 
  (2^(2/3)*x*HypergeometricPFQ[{3/2}, {4/3, 8/3}, 
 (-2*a*x^3)/9])/(3^(1/3)*a^(1/3)) - 
  (3*3^(2/3)*a^(2/3)*x^4*HypergeometricPFQ[{5/2}, {7/3, 11/3}, 
 (-2*a*x^3)/9])/(80*2^(1/3))) + 
 C[3]*((-16*2^(1/3)*HypergeometricPFQ[{7/6}, {5/3, 7/3}, 
 (-2*a*x^3)/9])/(9*3^(2/3)*a^(2/3)) - 
  (2*2^(1/3)*a^(1/3)*x^3*HypergeometricPFQ[{7/6}, {5/3, 7/3}, 
 (-2*a*x^3)/9])/(9*3^(2/3)) + 
   (2*2^(1/3)*a^(1/3)*x^3*HypergeometricPFQ[{13/6}, {8/3, 10/3}, 
  (-2*a*x^3)/9])/(5*3^(2/3)) - 
  (13*a^(4/3)*x^6*HypergeometricPFQ[{19/6}, {11/3, 13/3}, 
 (-2*a*x^3)/9])/(900*6^(2/3)))}}
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  • $\begingroup$ Thank you so much! $\endgroup$
    – Gabri
    May 9 '21 at 15:28

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