Solving x²y'' + xy' + (x² - 1/4)y = 0

Differential Equations · Axiom Academy

Apply the Frobenius method to the Bessel equation of order — a special case whose series solutions collapse to elementary functions. Find the general solution near the regular singular point x = 0 of — the Bessel equation of order . Nice work! You solved the Bessel equation of order with the Frobenius method — and saw that its series solutions are actually elementary functions. Indicial equation: comes from the coefficient of the lowest power x^r , giving roots . Roots differ by an integer ( r_1 - r_2 = 1 ), which in GENERAL can force a log term into the second solution — but not always: here the recurrence coefficient that would need to blow up is automatically zero, so BOTH solutions stay clean series with no . General solution: y = c_1 y_1(x) + c_2 y_2(x) , where y_1 corresponds to and y_2 to . The punchline — order is elementary: summing each series in closed form gives and — ordinary sine and cosine, just rescaled. Most Bessel functions do NOT reduce this way; half-integer order is the special exception. The Frobenius method is the standard technique for any 2nd-order linear ODE near a regular singular point — the same four-step pattern (assume, substitute, extract the indicial equation, solve) applies to every Bessel equation and many other equations in physics and engineering.

This is the written version of the interactive lesson above. See the full Differential Equations course.