# MMA breaks down when DSolve -ing a third-order linear ODE

Bug introduced in 9.0 or earlier and fixed in 10.4.0

My code is:

DSolve[y'''[x] + (x^3 + a^3) y[x] == 0, y[x], x] It should be noted that similar code not only meets the similar problem but also returns more warning messages:

DSolve[y'''[x] + (x^3 + a) y[x] == 0, y[x], x] • Bug reported internally. Thank you! Jul 22 '15 at 15:07
• Confirmed on V9 too Jul 22 '15 at 18:11
• Have you tried using DifferentialRoot[] for this ODE? Jul 22 '15 at 19:06
• @J.M. The code FunctionExpand[ DifferentialRoot[ Function[{y, x}, {y'''[x] + (x^3 + a^3) y[x] == 0, y == 0, y' == 1, y'' == 1}]][x]] meets the same problem. Jul 23 '15 at 2:57
• Okay, I'd guess that what you've seen is related to what's happening within DSolve[]. But if you use the DifferentialRoot[] as is, you can still evaluate it at numerical values, yes? Jul 23 '15 at 3:02

This bug has been fixed as of Mathematica 10.4.0.

DSolve[y'''[x] + (x^3 + a^3) y[x] == 0, y[x], x]

(* {{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]},
{(\[FormalX]^3 + a^3)*\[FormalY][\[FormalX]] +
Derivative[\[FormalY]][\[FormalX]] == 0, \[FormalY] ==
C,
Derivative[\[FormalY]] == C,
Derivative[\[FormalY]] == C}]][x]}} *)


Interested to know how this DE arises. Note that you can also use ParametricNDSolve

psol = ParametricNDSolve[{Derivative[y][x] + (x^3 + a^3) y[x] == 0,
y == b, y' == c, y'' == d}, y, {x, -3, 3}, {a, b, c, d}]


and visualize:

Manipulate[Plot[(y[a, b, c, d] /. psol)[x], {x, -3, 3}, PlotRange -> 5],
{a, -1, 1}, {{b, 1}, -1, 1}, {{c, 0}, -1, 1}, {{d, 0}, -1, 1}]


which is much faster than DifferentialRoot.

When $a=0$, exact solution yields 3 hypergeometric functions. The 3 linearly independent solutions are not hypergeometric otherwise. Series solution about $x=0$, or $x=-a$, could be useful, depending upon the parameter range of interest.

• I am interested in LaplaceTransform-invariant functions and the equation I posed above is just one class of the third-order equations that yields them. Aug 5 '15 at 11:10
• I don't see how solutions to these equations, which appear to be unbounded for large Abs[x], could be LaplaceTransform-invariant functions. Do you have a reference for this? Aug 7 '15 at 5:06
• The boundary condition is given arbitarily to test whether mathematica can solve it,so it doesn't related to my comment above. In addition,it's not hard to find that Weber function is LaplaceTransform-invariant in physical sense. Aug 7 '15 at 7:20
• I cannot see how the Weber function is LaplaceTransform-invariant, and I don't know what you mean about "in the physical sense". In the "mathematical sense" I played around in some special cases, and I looked in Tables of Laplace Transforms by Oberhettinger and Badii, which includes transforms of Weber functions, but your assertion is still not clear to me. Aug 7 '15 at 8:03