Solving a difficult differential equation

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Juseb3
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in attachment you can see the equation
 
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I'm sorry, I know that if Ed has a solution, but I really need help.
 
It is a first order linear differential equation, but the coefficients are not constant.

It's solution isn't trivital; for me.

I am student of undergraduate of civil engineering.

The Ed is of a problem of a tapered beam
 
Juseb3 said:
It is a first order linear differential equation, but the coefficients are not constant.

It's solution isn't trivital; for me.

I am student of undergraduate of civil engineering.

The Ed is of a problem of a tapered beam

Actually, it's a second-order one.
 
It is easier to see how to proceed if you write the constants on a simpler form.
 
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  • Handwritten-style derivation: a differential equation with substitutions a = r/L, b = P/EI, c = (Ma(1+L)+Mb)/EI, leading to (1+ax)^n u'' - bu = c, then u = U - c/b, x = (X-1)/a, B = b/a^2, ending with X^n U'' - BU = 0, described as an easily solvable ODE with Bessel functions.
    Bessel ODE.JPG
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For solving (X^n)*U''-B*U=0 :
 
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  • Typeset equations: solutions of x^(2-1/ν) d²y/dx² - λ²x y = 0 are y = c1 sqrt(x) J_ν(θ) + c2 sqrt(x) J_{-ν}(θ), where θ = 2λν x^{1/(2ν)}.
    Bessel.JPG
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