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# Differential Equations/Non-constant 2nd order differential eqn

Question
Hello Bob,

I am currently working on the following equation. I have tried quite a few tricks to solve the problem but can not find an analytic solution. In the case where L=0, I do find an analytic solution.

Here is the equation:

(R+L)^2*D[(R+L)^2*D[f(X),X],X]-A^2*(1+L/R)^3*f(X)=0
with R=b*Sqrt[1-X^2], L>0, b>0, A real

With L=0 I have a soution as I said earlier, but also when I take out the term L/R in (1+L/R)^3. I feel that this exact term is making the equation difficult to solve, so I am trying ways to bypass that, but did not come up with anything good. I also tried linearizing the terms by supposing    L << R but it does not do me any good.

Do you have any advice?

Best regards,

Benoit

Benoit,

Try writing the whole ode out as:-

y'' +ay' + by = 0   (1)

Put y=f(x)u(x),   (2), f and u ti be determined.

then y'=f'u + fu'

so y''=f''u + 2f'u' + fu''

so (1) => f''u + 2f'u' + fu'' + a(f'u + fu') + bfu = 0

so fu'' + (2f' + af)u' + c (=other terms)u = 0

Now choose an f to kill the u' term, so we solve 2f' + af = 0,

the solution of this being f(x) = exp (- \int a(x)dx).

Hence we now just need to solve fu'' + cu = 0,

which we can write as u'' + du = 0, where d = c/f,

and we use the same method, so that u(x) = exp (- \int d(x)dx),

and those both go into (2) to give y.

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Differential Equations

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#### Bob Marlow

##### Expertise

Questions on partial differential equations, particularly solution of time-dependent ones, and particularly parabolic. I have some knowledge of ODEs, but am out of practice with them.

##### Experience

My thesis was in this area.

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