Differential operators are a noncommutative domain with operations such as addition, multiplication, application, and so on. A differential operator  in  is an expression , where  are elements of C(x). An element  in  corresponds to a linear homogeneous differential equation .

Multiplication (see DEtools[mult]) in the ring  corresponds to composition of differential operators. So if , then . In particular, .

As examples of the types of algebraic operations you can do, consider the three differential operators:

You can multiply these operators and note that multiplication is noncommutative:

The argument  tells the mult command that the multiplication is over the differential domain specified by  and , and so the variables a and b are constants. This can also be set by the environment variable , if you expect to use the same domain throughout.

The concept of a one-sided lowest common multiple and greatest common divisor exists in such domains. For example,

and

One can check this by using right or left division. In this case, you have

which in both cases gives a quotient with 0 remainder. You can check the computation by multiplying:

The section contains subsections on closed-form solutions, and on other factorization options.

One can take advantage of the differential operator tools in DEtools to construct solutions of differential equations from known equations. For example,

is a differential operator with Bessel function solutions. Indeed we have

The third symmetric power of  then generates a differential equation that has the power combinations of the above as its solutions. For example,

We can construct products of solutions using the symmetric_product function. For example,

produces an exponential solution. In this case

gives a solution that is the product of solving df and df2.

To create an equation having an exponential solution, we can either explicitly construct operators from exponential equations, or we can use the exterior power operation. If an operator L has a right factor of order n, then the n-th exterior power of L has a right factor of order 1. For example

shows this right-hand factor. This can also be seen with