Non-commutativity
It is natural to think about extending this method to solve differential equations with variable coefficients. Some readers might have tried to manipulate it themselves but felt confused by not being able to obtain a suitable solution. This is where the non-Abelian nature of the group manifests itself. First, let us use an example to illustrate this. We consider the composition of operators: (D-x)(D+x) = D^2 - x^2 + (Dx - xD)
We must be cautious when using the commutative law. We denote [P,Q] = PQ - QP, where P and Q are two operators. This is the "commutator" in quantum mechanics, used to measure the degree of commutativity between operator P and operator Q. Of course, it is itself an operator. Let us first find out what [D,x] gives (if it were 0, it would indicate that the operations are commutative). What exactly does it equal? It is not obvious at first glance, so we act it upon a function: [D,x]y = (Dx - xD)y = D(xy) - xDy = yDx + xDy - xDy = y
Since the operator closest to the function acts first, Dxy means x acts on y first, and then D acts on (xy); whereas xDy means D acts on y first, and then x acts on Dy. Ultimately, we obtain: [D,x]y = y = 1 \times y
That is to say, [D,x] acting on y is equivalent to multiplying y by 1, so [D,x] = 1
Thus, (D-x)(D+x) = D^2 - x^2 + 1.
In this way, the general solution for a class of second-order variable coefficient equations D^2 y - (x^2 - 1)y = f(x) is: y = (D+x)^{-1} (D-x)^{-1} f(x)
According to equation (3) from the previous section, the specific solution can be found.
Linear Ordinary Differential Equations with Variable Coefficients
In theory, general linear differential equations with variable coefficients can also be solved using similar decomposition methods. However, unfortunately, due to the non-Abelian nature of the operators, it is difficult to determine the specific functional forms after decomposition. Because the differentiation operator and constants are commutative, one can directly decompose constant coefficient differential equations by finding roots; but as shown in the example above, for a mixture of general functions and the differentiation operator, root-finding becomes unfeasible due to non-commutativity.
Consider a second-order linear ordinary differential equation: y = f(x) \tag{12} And consider the operator: [D - a(x)][D + a(x)] = D^2 - a^2(x) + a'(x)
We hope to find an a(x) that satisfies: -a^2(x) + a'(x) = g(x) \tag{13} If so, we can successfully decompose (12). Moreover, we only need one particular solution of (13). Equations of the form (13) are called Riccati equations. Although it is first-order, it is essentially still a second-order linear ordinary differential equation with variable coefficients. By letting a(x) = -\frac{u'}{u}, we have: a'(x) = -\frac{u''}{u} + \left(\frac{u'}{u}\right)^2 = -\frac{u''}{u} + a^2(x)
Thus (13) can be transformed into (14): u'' + g(x)u = 0 \tag{14} This is again a second-order linear ordinary differential equation with variable coefficients, so it feels like falling into an infinite loop. Therefore, the generalizability of this method is not great. However, it can be used for "retrospective analysis"—that is, first specifying the operator, then determining a specific equation form, and then using this operator to solve that specific class of equations. This is often a feasible approach; for instance, the harmonic oscillator problem in quantum mechanics is handled using a similar method.
We can also find that knowing a particular solution to (14) is equivalent to knowing the general solution to (12). In other words: a particular solution to [D^2 + g(x)]y = 0 is equivalent to the general solution to [D^2 + g(x)]y = f(x). This cannot but make us feel the wonder of mathematics: local properties can determine global properties. Broadly speaking, can this also be seen as a kind of "analogy"? This reminds me of a striking similarity in quantum mechanics: similarly, if the initial form of a wave function is known, its general form can be obtained through Fourier analysis. Mathematics always reflects its unexpected beauty in various places.
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