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Can the given equation be transformed to the Bessel equation?. If yes, how it can be transformed then?.
Y''(x)+a*Y'(x)+b^(2)*e^(2ax)Y(x)=0

7444_S
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  • Welcome to the Mathematica Stack Exchange. If you are just starting out with Mathematica, please check out the book written by the inventor. Please indicate (1) if this is a Mathematica question (2) and if so, please include the code that you have tried out so far. The idea is to include enough code to replicate your problem so that respondents can help you in a focused manner. It is called a minimal working example. Your question, as written, seems like a math question suitable for the Mathematics SE. – Syed Feb 22 '22 at 04:34
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    This seems to be a question for the https://math.stackexchange.com/ –  Feb 22 '22 at 04:59
  • the exp(2 a x) makes it not possible to apply the standard method to do this transformation. Are you sure this term is there in the original ode? – Nasser Feb 22 '22 at 05:32
  • Nasser...yes sir, the exponential term is present in the ODE. BTW, a and b are constant. – 7444_S Feb 22 '22 at 05:40
  • My question is similar to the one you responded...https://mathematica.stackexchange.com/questions/34423/numerical-solution-of-bessel-like-equation-using-ndsolve?newreg=b0ecc340cb284180bc1fb91c547b0b22 – 7444_S Feb 22 '22 at 06:21
  • There is standard method to convert second order ode to Bessel form ode. But it does not work when the coefficients depend on $x$ as you have. At least I do not know how to make it work now. But why do you want to convert the ode to Bessel form? Mathematica DSolve works on it as is and solves it. – Nasser Feb 22 '22 at 06:26
  • Mathematica gives bessel function j and y of first kind for this equation. Looking at the equation, its form does not match with that of standard bessel equation. How to derive the solution of this equation based on theoretical approach as its does not match bessel equation. – 7444_S Feb 22 '22 at 06:37
  • "Mathematica gives bessel function j and y of first kind for this equation." Do you mean DSolve does? It does not for me. Can you show what you tried and the code? On my system V 13.01 it gives this

    Mathematica graphics But if you want to transform the ode to Bessel form first. then may be asking at the math form would be better in this case as suggested above.

     – Nasser Feb 22 '22 at 06:45

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The only way I can see is to let a->1/x

ode = Y''[x] + a Y'[x] + b^2 E^(2 a x) Y[x] == 0 /. a -> 1/x
(*  E^2 b^2 Y[x] + Y''[x] + Y'[x]/x == 0  *)

and we get Bessel's eqn of order zero.

DSolve[ode, Y[x], x]
{{Y[x] -> C[1] BesselJ[0, b E x] + C[2] BesselY[0, b E x]}}

If a is to remain a constant, this, of course, won't work.

Bill Watts
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