Logarithmic Cases

Differential Equations · Axiom Academy

When indicial roots repeat or differ by an integer, the standard Frobenius series isn't enough — a ln(x) term completes the picture. 1. Three Cases, One Question: What Is r ₁ − r ₂? Every regular singular point gives an indicial equation F(r)=0 with two roots . Which method you use for the second solution y_2 depends entirely on the gap between them . The Frobenius ansatz — valid for the larger root r_1 in every case Two clean series. No log, ever. Conditional. Log needed only if the recursion breaks. 2. Repeated Root: Why the Log Is Forced When r_1=r_2=r , the Frobenius method hands you exactly one series solution y_1 . To find a genuinely independent y_2 , apply reduction of order : write y_2=y_1v(x) and solve for v . Abel's formula gives the Wronskian for a repeated root. Since v'=W/y_1^2 and , the leading behavior is — and . 3. Integer Difference: A Recursion That Might Break When r_1-r_2=N (a positive integer), the larger root r_1 always gives a valid series. For the smaller root r_2 , building the coefficients a_n by substitution hits a decisive moment right at n=N . Always works — the large root never has this problem C is determined by the obstruction check above — it can genuinely be zero. Always test the recursion before assuming a log is needed. 4. Worked Example: Bessel's Equation of Order Zero Bessel's equation with is the textbook case of a repeated indicial root, and it's where the second solution Y_0 was first constructed this way.

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