The Coupled Differential System

Abstract

We describe in this chapter the solution to the coupled differential system problem, i.e. given a differential field K of characteristic 0 and f ₁, f ₂, g ₁, g ₂ in K, to decide whether the system of equations

$$(_{D{y_2}}^{D{y_1}}) + (_{{f_2}}^{{f_1}}{}_{{f_1}}^{ - {f_2}})(_{{y_2}}^{{y_1}}) = (_{{g_2}}^{{g_1}})$$

(8.1)

has a solution in K × K, and to find one if there are some. It turns out that (8.1) is not really a second order equation, but the coupled system for the real and imaginary parts of a Risch differential equation. Indeed, suppose that (y ₁, y ₂) ∈ K × K is a solution of the slightly more general system

$$(_{D{y_2}}^{D{y_1}}) + (_{{f_2}}^{{f_1}}{}_{{f_1}}^{a{f_2}})(_{{y_2}}^{{y_1}}) = (_{{g_2}}^{{g_1}})$$

(8.2)

for an arbitrary a ∈ Const _D (K). Then, since $D\sqrt a = 0$ by Lemma 3.3.2, writing $y = {y_1} + {y_2}\sqrt a$ we have

$$\begin{gathered} Dy + \left( {f_1 + f_2 \sqrt a } \right)y = Dy_1 + D\left( {y_2 } \right)\sqrt a + \left( {f_1 + f_2 \sqrt a } \right)\left( {y_1 + y_2 \sqrt a } \right) \hfill \\ = Dy_1 + f_1 y_1 + af_2 y_2 + \left( {Dy_2 + f_2 y_1 + f_1 y_2 } \right)\sqrt a \hfill \\ = g_1 + g_2 \sqrt a \hfill \\ \end{gathered} $$

which implies that y is a solution in $K\sqrt a$ of the Risch differential equation

$$Dy + ({f_1} + {f_2}\left){\vphantom{1a}}\right. \!\!\!\!\overline{\,\,\,\vphantom 1{a}})y = {g_1} + {g_2}\sqrt a $$

(8.3)

.