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differential equations - DSolve gives complex function although the solution is a real one


I have a problem with the DSolve[] command in mathematica 8. Solving the the following 4th order differential equation spits out a complex solution although it should be a real one. The equation is:


y''''[x] + a y[x] == 0

Solving this equation by hand yields a solution with only real parts. All constants and boundary conditions are also real numbers.


The solution I get by hand is:


y1[x_] = (C[5] E^(Power[a, (4)^-1]/Power[2, (2)^-1] x) + 
C[6] E^(-(Power[a, (4)^-1]/Power[2, (2)^-1]) x)) Cos[
Power[a, (4)^-1]/Power[2, (2)^-1]
x] + (C[7] E^(Power[a, (4)^-1]/Power[2, (2)^-1] x) + 

C[8] E^(-(Power[a, (4)^-1]/Power[2, (2)^-1]) x)) Sin[
Power[a, (4)^-1]/Power[2, (2)^-1] x];

Now I have to solve for the constants C[5]...C[8]. This arises a similar issue. I use the Solve[] command with the boundary conditions


Solve[{y1''[-c] == ic0, y1''[c] == ic0 , y1'''[-c] == ic1 , 
y1'''[c] == - ic1 }, {C[5], C[6], C[7], C[8]} ];

The constants C[5]...C[8] are now real if using //Simplify and complex if using //FullSimplify.


Any idea what the reasons are? The notebook with my calculations can be downloaded under: http://dl.dropbox.com/u/4920002/DGL_4th_Order_with_own_solution.nb


In further work I have to use DSolve[] and I would like to understand the issue here.




Answer



Here is the reply I got from the Mathematica support team:



Thank you for your prompt reply.


Please find attached a small notebook which demonstrates using FullSimplify directly on the output of your first DSolve expression to obtain a symbolic real-valued formula.


The formula itself does contain imaginary values, but these zero each other out as seen when the formula is evaluated numerically. Unfortunately, the corresponding "hidden zero" in the imaginary component cannot be eliminated by either of two typical techniques:


(1) increasing the internal precision available to the N function when evaluating the formula numerically, or


(2) using FullSimplify which is sometimes successful in removing hidden zeros from symbolic formulas.


The recommended approach in cases like this is to use the Chop function, which will eliminate near-zero numerical values in any given expression.


For more information, please see the Possible Issues subsection of the $MaxExtraPrecision page.



For details on the Chop function and how its behavior can be customized, please see its corresponding Documentation Center page.


Please let me know if you have any questions. Thank you.


{sol} = DSolve[{y''''[x] + a y[x] == 0, y''[-c] == ic0, y''[c] == ic0,
 y'''[-c] == ic1, y'''[c] == -ic1}, y[x], x];

val = FullSimplify[sol, a > 0 && {ic0, ic1, c, x} \[Element] Reals]

N[val]
% // Chop


Block[{$MaxExtraPrecision = 10000}, N[val]]
% // Chop

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