4.2 Reduction to Sturm–Liouville Form
A second-order linear equation need not arrive in the form (4.1), and none of the classical equations does. Any such equation can be put into that form, and the procedure is short.
Theorem 4.2.1 (Reduction to self-adjoint form). The equation \[a_2(x)\,y'' + a_1(x)\,y' + \left [a_0(x)+\lambda \right ]y = 0, \qquad a_2(x)\neq 0,\] becomes a Sturm–Liouville equation on multiplication by the integrating factor \[\mu (x) = \frac {1}{a_2(x)}\exp \left (\int ^{x}\frac {a_1(t)}{a_2(t)}\,dt\right ),\] with \[p(x) = \exp \left (\int ^{x}\frac {a_1(t)}{a_2(t)}\,dt\right ), \qquad q(x) = a_0(x)\,\mu (x), \qquad w(x) = \mu (x).\]
Proof. Multiplying through by \(\mu \) gives \(\mu a_2 y'' + \mu a_1 y' + \mu (a_0+\lambda )y = 0\). For the first two terms to be \(\left (py'\right )' = py''+p'y'\) we need \(\mu a_2 = p\) and \(\mu a_1 = p'\). Dividing, \(p'/p = a_1/a_2\), whence \(p = \exp \left (\int a_1/a_2\right )\) and \(\mu = p/a_2\), which is the stated factor. The remaining term is \(\mu (a_0+\lambda )y = \left (q+\lambda w\right )y\) with \(q=a_0\mu \) and \(w=\mu \). □
Note 4.2.2. Theorem 4.2.1 is where the weight function comes from. It is not selected to make an orthogonality relation come out; it is the integrating factor \(\mu = p/a_2\) forced by the equation itself. This is the answer to the question Chapter 3 left open, and the calculations of the next section simply carry it out for each family in turn.
Example 4.2.3 (Legendre). Legendre’s equation is \[\left (1-x^{2}\right )y'' - 2x\,y' + n(n+1)\,y = 0 .\] Here \(a_2 = 1-x^{2}\) and \(a_1 = -2x\), so \(a_1/a_2 = -2x/(1-x^{2})\) and \[\int \frac {-2x}{1-x^{2}}\,dx = \ln \left (1-x^{2}\right ), \qquad p(x) = 1-x^{2},\qquad \mu = \frac {p}{a_2} = 1 .\] The weight is therefore \(w(x)=1\) and the equation is already self-adjoint: \[\left [\left (1-x^{2}\right )y'\right ]' + n(n+1)\,y = 0 .\] This is why the Legendre polynomials are orthogonal on \([-1,1]\) with weight \(1\), and it required no choice at all.
Example 4.2.4 (Hermite). The notes use the probabilists’ Hermite polynomials of Section 3.6, which satisfy \[y'' - x\,y' + n\,y = 0 .\] Here \(a_2=1\) and \(a_1=-x\), so \(\int (-x)\,dx = -x^{2}/2\) and \(p(x) = w(x) = e^{-x^{2}/2}\), giving \[\left (e^{-x^{2}/2}\,y'\right )' + n\,e^{-x^{2}/2}\,y = 0 .\] The weight \(e^{-x^{2}/2}\) in the table of Section 3.1 is precisely this integrating factor.
Remark 4.2.5. Most texts define Hermite polynomials by \(H_n(x)=(-1)^n e^{x^{2}}\frac {d^n}{dx^n}e^{-x^{2}}\), the physicists’ convention, whose equation is \(y''-2xy'+2ny=0\) and whose weight is \(e^{-x^{2}}\) with eigenvalue \(2n\). These notes use the probabilists’ convention throughout, with weight \(e^{-x^{2}/2}\) and eigenvalue \(n\); the two are related by \(H_n^{\text {phys}}(x)=2^{n/2}H_n^{\text {prob}}(x\sqrt 2)\). Neither is more correct, but a reader comparing these notes with another book should check which is in use before comparing formulae.
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