The Risch Differential Equation

Abstract

We describe in this chapter the solution to the Risch differential equation problem, i.e. given a differential field K of characteristic 0 and f, g ∈ K, to decide whether the equation

$${\text{Dy + fy = g}}$$

(6.1)

has a solution in K, and to find one if there are some. We only study equation (6.1) in the transcendental case, i.e. when K is a simple monomial extension of a differential subfield k, so for the rest of this chapter, let k be a differential field of characteristic 0 and t be a monomial over k. We suppose that the coefficients f and g of our equation are in k(t) and look for a solution y ∈ k(t). The algorithm we present in this chapter proceeds as follows:

• 1. Compute the normal part of the denominator of any solution. This reduces the problem to finding a solution in k<t>.

• 2. Compute the special part of the denominator of any solution. This reduces the problem to finding a solution in k[t].

• 3. Bound the degree of any solution in k[t].

• 4. Reduce equation (6.1) to one of a similar form but with f, g ∈ k[t].

• 5. Find the solutions in k[t] of bounded degree of the reduced equation.