= Frobenius case classification
{c}
{title2=$\Delta=r_1-r_2$}
For a real pair of <indicial roots> $r_1\geq r_2$ at a <regular singular point> of a <second-order linear differential equation>, the second local solution is classified by the gap $\Delta$ and by the <recurrence relation> for the smaller <indicial root>. Case I has $\Delta\notin\mathbb Z$ and two pure <Frobenius solutions>. Case II has $\Delta=0$ and an independent <Logarithmic Frobenius solution>. With a positive integer gap, Case IIIa means the exceptional coefficient equation is inconsistent, so a nonzero multiple of the first <Frobenius solution> times a <logarithm> is necessary. Case IIIb means that equation is consistent and leaves a free coefficient, so there are two pure <Frobenius solutions>. This IIIa/IIIb convention is the one used in the Oxford course notes.https://www.cjhb.site/Files.php/Books/Differential%20Equation/02_ODE/Differential%20Equations%20II.pdf#page=55[Oxford A6 Differential Equations II lecture notes, draft 3 January 2023, §5.2.5, printed p. 5–7: III(a) has a nonzero right-hand side at the exceptional index; III(b) has a zero right-hand side]{ref}
The free coefficient in Case IIIb adds a multiple of the first <Frobenius solution>, as explained by the <undetermined coefficient at Frobenius resonance>. Stating whether the <logarithm> occurs is useful alongside the case name because other courses may choose different labels.
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