Bessel Equation Solutions

Equation: x²y″ + xy′ + (x² − n²)y = 0  —  select a function family from the tabs below

Definition
Jₙ(x) is the standard Bessel function of the first kind. It is finite and well-behaved at x = 0.
Behaviour
Oscillates like a damped sine/cosine. Amplitude decreases as x grows. Higher n shifts first zero right.
Applications
Circular drum vibrations, heat conduction in cylinders, EM waveguides, optics, signal processing.
Key property
J₀(0) = 1, Jₙ(0) = 0 for n>0. Infinite zeros distributed along positive x-axis.
0→15
500 pts
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Definition
Yₙ(x) is the Bessel function of the second kind (Neumann function). It diverges at x = 0.
Behaviour
Oscillates like Jₙ(x) for large x. Singular (goes to −∞) as x→0. Out-of-phase with Jₙ.
Applications
Used when origin is excluded (hollow cylinders, exterior regions). Pairs with Jₙ as independent solution.
Key property
Yₙ(x) → −∞ as x→0. General solution to Bessel eq: c₁Jₙ + c₂Yₙ.
0→15
500 pts
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Definition
Iₙ(x) is the modified Bessel function of the first kind. Comes from replacing x with ix in Bessel's equation.
Behaviour
Grows exponentially for large x. No oscillation. I₀(0) = 1, Iₙ(0) = 0 for n>0.
Applications
Heat transfer in cylinders, antenna theory, statistics (von Mises distribution), signal processing.
Key property
Iₙ(x) ~ eˣ / √(2πx) for large x. Always positive for x>0. Real-valued for all real x.
0→8
500 pts
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Definition
Kₙ(x) is the modified Bessel function of the second kind. Complements Iₙ(x) as independent solution.
Behaviour
Decays exponentially to zero for large x. Singular at x = 0. Always positive for x>0.
Applications
Exterior problems (outside a cylinder), nuclear physics, quantum field theory, telecommunications.
Key property
Kₙ(x) ~ √(π/2x) e⁻ˣ for large x. General modified solution: c₁Iₙ + c₂Kₙ.
0→8
500 pts
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