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I wanted to find the general solution of the following ODE. The ODE is not a linear differential with constant coefficient. Its is the fourth order differential equation. I used DSolve function along with boundary conditions, but unable to solve. How to solve this. In the paper I am referring, the general solution is given by Bessel functions and modified Bessel functions of second order. Following is the Mathematica code that I tried to obtain the general solution, But I am getting something which is called as MeijerG. How to convert this function to Bessel function?

\begin{align*} \frac{d^2}{dx^2}\left(\left(1-\frac{a}{b}x\right)^4\frac{d^2Y(x)}{dx^2}\right)-\frac{\omega^2}{c b^2}\left(1-\frac{a}{b}x\right)^2 Y(x) \end{align*}

    s1 = (1 - a/b*x)^4*D[Y[x], {x, 2}];
    a1 = D[s1, {x, 2}];
    a2 = ω^2/(c*b^2) (1 - a/b*x)^2*Y[x];
    e1 = Simplify[a1 - a2]
    FullSimplify[DSolve[e1 == 0, Y[x], x]](*General soluction*)
    FullSimplify[DSolve[{e1 == 0, Y[0] == 0, Y''[0] == 0, Y[1] == 0, Y''[1] ==0},Y[x], x]]

The Link of the journal paper is here link to journal

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  • $\begingroup$ What exactly is your question? Find the analytic particular solution? Make Mathematica express the general solution in the same way as in the paper? Is the particular solution given in that paper? Can you add the link for the paper? $\endgroup$ – xzczd Nov 26 '18 at 13:49
  • $\begingroup$ I actually looking for the analytical solution of the differential equation. In the paper the have mentioned the analytical solution. I just wanted to reproduce the same results. journals.sagepub.com/doi/abs/10.1177/1077546314550699 $\endgroup$ – acoustics Nov 26 '18 at 16:23
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    $\begingroup$ If I understand the paper correctly, the so-called analytic solution in the paper still involves a unsolved transcendental equation. (At the end of Section 2.1: "…the natural frequencies of free bending vibrations for the beam with variable cross section are obtained by solving numerically a transcendental equation derived by employing the boundary conditions at the beam-ends. ") So, I doubt if the solution can be expressed in an explicit way. $\endgroup$ – xzczd Nov 27 '18 at 15:57
  • $\begingroup$ Yes your right, My interest is to get the general solution mentioned in the paper using Mathematica. If I can able to get the general solution, then I apply four boundary conditions on that, I will get four system of equations. If I take the determinant of the matrix containing these systems of equations, I will get the transcendental equation. Later I will solve this numerically using NSolve in Mathematica. Is it possible to do in Mathematica. Because when I solved for general solution I am getting some weird results showing some thing like "MeijerG" I dont know what is this? $\endgroup$ – acoustics Nov 27 '18 at 16:13
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    $\begingroup$ This is a special function. For more information you may check the document of it and the corresponding wikipedia page. So the question now boils down to "how to express the MeijerG as Bessel function?". BTW you should clarify your question by adding these information to the body of the question, not only in the comment. $\endgroup$ – xzczd Nov 27 '18 at 16:23
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We make obvious simplifications.

(*b=1/d^2*Sqrt[(Iyy*Y)/(ρ*A)];k1=α/d;k2=ω^2/(b*d^4)*)
(*-1+k1*x->x;k=k2*k1^4;*)
a = x^4*D[W[x], {x, 2}];
a1 = D[a, {x, 2}];

a2 = k*x^2*W[x];
e1 = Simplify[a1 - a2]
(*x^2 (-k W[x] + 12 W'[x] + x (8 W'''[x] + x W''''[x]))*)

Then the basic equation has the form.

eq=-k W[x] + 12 W'[x] + x (8 W'''[x] + x W''''[x]);

General solution of the equation

DSolve[eq == 0, W[x], x]

(*{{W[x] -> -((
     3 I (BesselI[2, 2 k^(1/4) Sqrt[x]] - 
        BesselJ[2, 2 k^(1/4) Sqrt[x]]) C[1])/(
     4 Sqrt[k]
       x)) - ((BesselI[2, 2 k^(1/4) Sqrt[x]] + 
       BesselJ[2, 2 k^(1/4) Sqrt[x]]) C[3])/(Sqrt[k] x) + 
    C[4] MeijerG[{{}, {}}, {{-1, 0}, {-(1/2), 1/2}}, (k x^2)/16] + 
    C[2] MeijerG[{{}, {}}, {{-(1/2), 1/2}, {-1, 0}}, (k x^2)/16]}}*)

To find a solution to an equation with boundary conditions, you can use General solution and Solve[]. Do not use FullSimplify[].

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