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I'm trying to solve a differential equation which solution is in the form of Bessel functions. One of the boundary conditions is at infinity. I use:

\[Psi][R_]ψ[R_] = \[Psi][R]ψ[R] /. DSolve[{\[Psi]''[R]ψ''[R] + (Derivative[1][\[Psi]][R]ψ'[R])/
R - \[Kappa]^2κ^2 \[Psi][R]ψ[R] == 0,
\[Psi][a                       ψ[a/2] ==
Psi0, \[Psi][M]ψ[M] == 0}, \[Psi][R]ψ[R], R][[1]]Limit[\[Psi][R]R][[1]]

Limit[ψ[R], M -> Infinity]Simplify[%Infinity]

Simplify[%, {r > 0 && \[Kappa]κ > 0 && a > 0 && Psi0 \[Element]∈ Reals}]


And obtain:

How can I obtain the solution of my problem overpassing the time limit?

I'm trying to solve a differential equation which solution is in the form of Bessel functions. One of the boundary conditions is at infinity. I use:

\[Psi][R_] = \[Psi][R] /. DSolve[{\[Psi]''[R] + (Derivative[1][\[Psi]][R])/
R - \[Kappa]^2 \[Psi][R] == 0, \[Psi][a/2] ==
Psi0, \[Psi][M] == 0}, \[Psi][R], R][[1]]Limit[\[Psi][R], M -> Infinity]Simplify[%, {r > 0 && \[Kappa] > 0 && a > 0 && Psi0 \[Element] Reals}]


And obtain:

How can I obtain the solution of my problem overpassing the time limit?

I'm trying to solve a differential equation which solution is in the form of Bessel functions. One of the boundary conditions is at infinity. I use:

ψ[R_] = ψ[R] /. DSolve[{ψ''[R] + (ψ'[R])/R - κ^2 ψ[R] == 0,
ψ[a/2] == Psi0, ψ[M] == 0}, ψ[R], R][[1]]

Limit[ψ[R], M -> Infinity]

Simplify[%, {r > 0 && κ > 0 && a > 0 && Psi0 ∈ Reals}]


And obtain:

How can I obtain the solution of my problem overpassing the time limit?

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# How to solve a Bessel differential equation with a boundary condition at infinity?

I'm trying to solve a differential equation which solution is in the form of Bessel functions. One of the boundary conditions is at infinity. I use:

\[Psi][R_] = \[Psi][R] /. DSolve[{\[Psi]''[R] + (Derivative[1][\[Psi]][R])/
R - \[Kappa]^2 \[Psi][R] == 0, \[Psi][a/2] ==
Psi0, \[Psi][M] == 0}, \[Psi][R], R][[1]]Limit[\[Psi][R], M -> Infinity]Simplify[%, {r > 0 && \[Kappa] > 0 && a > 0 && Psi0 \[Element] Reals}]


And obtain:

How can I obtain the solution of my problem overpassing the time limit?