# Boundary value problems for fourth order ordinary differential equations

I am trying to solve a set of equations which are fourth order equations. I want to get the value of w

1/r ((-0.2409850746268657 +
270.59999999999997 r^2 ω^2) ψ[r] +
r (0.08050746268656715 Derivative[1][ψ][r] +
r (0.013388059701492531 (ψ^′′)[r] +
r (-0.01791044776119403
\!$$\*SuperscriptBox[\(ψ$$,
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction->None]\)[r] - 0.002238805970149254 r
\!$$\*SuperscriptBox[\(ψ$$,
TagBox[
RowBox[{"(", "4", ")"}],
Derivative],
MultilineFunction->None]\)[r])))) == 0
ψ[0.4] == 0
Derivative[1][ψ][0.4] == 0
-0.34 n^2 ψ[1] +
0.34 Derivative[1][ψ][1] + (ψ^′′)[1] == 0
9 (2.66 ψ[1] - 1.66 Derivative[1][ψ][1]) -
1. Derivative[1][ψ][1] + 1. (ψ^′′)[1] + 1.
\!$$\*SuperscriptBox[\(ψ$$,
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction->None]\)[1] == 0


It's too hard to me to solve; I hope someone can help me.Thanks in advance!

NDSolve does not recognize (ψ^′′)[r], by which I assume you mean ψ''[r]. Additionally, the quantities n and ω are undefined, so I set them both to 1. With these changes,

eq = {((-0.2409850746268657 + 276.03905999999995*r^2)*ψ[r] +
r*(0.08050746268656715*Derivative[1][ψ][r] +
r*(0.013388059701492531*Derivative[2][ψ][r] +
r*(-0.01791044776119403*Derivative[3][ψ][r] -
0.002238805970149254*r*Derivative[4][ψ][r]))))/r == 0,
ψ[0.4] == 0, Derivative[1][ψ][0.4] == 0,
-0.34*ψ[1] + 0.34*Derivative[1][ψ][1] + Derivative[2][ψ][1] == 0,
9*(2.66*ψ[1] - 1.66*Derivative[1][ψ][1]) - 1.*Derivative[1][ψ][1] +
1.*Derivative[2][ψ][1] + 1.*Derivative[3][ψ][1] == 0};
s = NDSolveValue[eq, ψ, {r, .4, 1}];
LogPlot[-s[r], {r, .4, 1}, PlotRange -> All, AxesLabel -> {r, ψ},
LabelStyle -> Directive[Black, Bold, 12]]


Other values for n and ω will, of course, produce different results. I hope this meet your needs.

• Thank you very much for your advices.I just copy the code and run it but failed,"NDSolveValue::bvluc:“ and it can't yield the plot Commented Jul 7, 2015 at 8:35
• At the same time,I try to slove the problem with bvp4c of Matlab. Commented Jul 7, 2015 at 8:38
• @1112131424 I just copied the code from my answer above, and it ran as before. I am using Mathematica 10.1. What version are you using? Commented Jul 7, 2015 at 11:57
• Mathematica 9.0 Commented Jul 7, 2015 at 12:04
• I just copied the code and run it with Mathematica 10.1,getting the same answer as you,I have been stuck in this problem for a long time,thank you! Commented Jul 7, 2015 at 12:29