Answer: x²d 2 y/dx 2 + xy_prime + (x²-n²)y = 0.
- A d<sup>2</sup>y/dx<sup>2</sup> + y = 0, the standard simple harmonic oscillator equation
- B x²d<sup>2</sup>y/dx<sup>2</sup> + xy_prime + (x²-n²)y = 0
- C d<sup>2</sup>y/dx<sup>2</sup> - y = 0, whose solutions are hyperbolic functions
- D d<sup>2</sup>y/dx<sup>2</sup> + xy = 0, a form resembling the Airy equation
Correct answer: B. x²d<sup>2</sup>y/dx<sup>2</sup> + xy_prime + (x²-n²)y = 0
Explanation: Bessel's equation: x²y''+xy'+(x²−n²)y=0, where n is the order. Solutions are Bessel functions of first kind Jₙ(x) and second kind Yₙ(x), expressed as power series. It arises in problems with cylindrical symmetry (wave equation in cylinders).
The general solution y=Ax² represents an entire FAMILY of curves, one for each value of the arbitrary constant A; a particular solution (fixed by an initial condition) selects exactly one curve from this family.
Concept context
Equations involving derivatives and their solutions