Boundary value problems for fourth order ordinary differential equations

Question

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!

Answer 1

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.