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I want to solve the following differential equation with mathematica:

 α β * w''''''[ξ] + ( 1 + α - p ) * w''''[ξ] + p/β * w''[ξ] = 0

The answer seems at first appearance really complicated.

w[ξ] -> 2 α β (-(( E^((Sqrt[-((-1 + p - α + Sqrt[ p^2 + 2 p (-1 + α) + (1 + α)^2])/(α \ β))] ξ)/Sqrt[2]) C[1])/(-1 + p - α + Sqrt[ p^2 + 2 p (-1 + α) + (1 + α)^2])) - ( E^(-((Sqrt[-((-1 + p - α + Sqrt[ p^2 + 2 p (-1 + α) + (1 + α)^2])/(α \ β))] ξ)/Sqrt[2])) C[2])/(-1 + p - α + Sqrt[ p^2 + 2 p (-1 + α) + (1 + α)^2]) + ( E^((Sqrt[( 1 - p + α + Sqrt[ p^2 + 2 p (-1 + α) + (1 + α)^2])/(α \ β)] ξ)/Sqrt[2]) C[3])/( 1 - p + α + Sqrt[ p^2 + 2 p (-1 + α) + (1 + α)^2]) + ( E^(-((Sqrt[( 1 - p + α + Sqrt[ p^2 + 2 p (-1 + α) + (1 + α)^2])/(α \ β)] ξ)/Sqrt[2])) C[4])/( 1 - p + α + Sqrt[ p^2 + 2 p (-1 + α) + (1 + α)^2])) + C[5] + ξ C[6]

So now I want to simplify the result above. I am sure that the result can be simplified with some own defined expressions like: 

a1 = Sqrt[p^2 + 2 p (-1 + α) + (1 + α)^2]

I have encountered similar differential equations with confusing solutions. I would really appreciate if someone could help me to solve this problem.

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    $\begingroup$ Probable duplicate: (3822) (please see the many links in my answer there) $\endgroup$ Commented Feb 3, 2015 at 10:58
  • $\begingroup$ Note that your original differential equation (which should use == rather than =) is actually a 4th order linear differential equation in v = w''. So you may want to try first simplifying the solution v[\[Xi]] of that 4th order equation before proceeding to integrate twice to come down to the original w. $\endgroup$ Commented Feb 3, 2015 at 15:06

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This is the right-hand part of your solution (taken from your post above):

 expr = 2 α β (-((E^((Sqrt[-((-1 + p - α + Sqrt[p^2 + 2 p (-1 + α) + (1 + \ α)^2])/(α β))] ξ)/Sqrt[2]) C[1])/(-1 + p - α + Sqrt[p^2 + 2 p (-1 + α) + (1 + α)^2])) - \ (E^(-((Sqrt[-((-1 + p - α + Sqrt[p^2 + 2 p (-1 + α) + (1 + \ α)^2])/(α β))] ξ)/Sqrt[2])) C[2])/(-1 + p - α + Sqrt[p^2 + 2 p (-1 + α) + (1 + α)^2]) + (E^((Sqrt[(1 - p + α + Sqrt[p^2 + 2 p (-1 + α) + (1 + \ α)^2])/(α β)] ξ)/Sqrt[2]) C[3])/(1 - p + α + Sqrt[p^2 + 2 p (-1 + α) + (1 + α)^2]) + (E^(-((Sqrt[(1 - p + α + Sqrt[p^2 + 2 p (-1 + α) + (1 + \ α)^2])/(α β)] ξ)/Sqrt[2])) C[4])/(1 - p + α + Sqrt[p^2 + 2 p (-1 + α) + (1 + α)^2])) + C[5] + ξ C[6];

Try the following:

expr /. p^2 + 2 p (-1 + α) + (1 + α)^2 -> a1^2 /. p -> 1 + Sqrt[a1^2] - b1 + α

where b1=1 + Sqrt[a1^2] - p + α. The result is somewhat more simple:

 2 α β (-(( E^((Sqrt[-((2 Sqrt[a1^2] - b1)/(α β))] ξ)/Sqrt[ 2]) C[1])/(2 Sqrt[a1^2] - b1)) - ( E^(-((Sqrt[-((2 Sqrt[a1^2] - b1)/(α β))] ξ)/Sqrt[ 2])) C[2])/(2 Sqrt[a1^2] - b1) + ( E^((Sqrt[b1/(α β)] ξ)/Sqrt[2]) C[3])/b1 + ( E^(-((Sqrt[b1/(α β)] ξ)/Sqrt[2])) C[4])/b1) + C[5] + ξ C[6]

It will look much more readable, when obtained in Mma.

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Have fun!

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