Nonalgebraic Ordinary Differential Equations

Let P= $P(y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots,v_{k})$be a polynomial in the variables $y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots,v_{k}$ with coefficients in a differential ring R of functions. So $P \in R\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k} \}$.

Let z_i= exp(Q_i) and let v_j= log(R_j), where Q_i= $Q_{i}(y,y^{(1)},\ldots,y^{(q_{i})})$ and R_j= $R_{j}(y,y^{(1)},\ldots,y^{(r_{j})})$ are polynomial functions with coefficients in the
differential ring R for all $i=1,\ldots,h$ and all $j=1,\ldots,k$, e.g. $Q_{i}, R_{j} \in R\{ y \}$ for all i and j. Moreover ord(Q_i)=q_i and ord(R_j)=r_j for all i and j.

The ordinary differential equation

$$\{P=P(y,y^{(1)},\ldots,y^{(n)},exp(Q_{1}),\ldots,exp(Q_{h}),log(R_{1}),\ldots, log(R_{k}))=0\}$$

is in general nonlinear and nonalgebraic.

Since exp(a) (respectively log(a)) with a in a differential ring Rare uniquely defined by the differential equations { y⁽¹⁾-a⁽¹⁾y=0 } (respectively { ay⁽¹⁾-a⁽¹⁾=0 }) up to a constant, then the equation $\{$P=0$\}$ is equivalent to the system

$(\alpha)$={ P= $P(y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots,v_{k})$=0, P⁽¹⁾=0,..., P^(m)=0, z_i= exp(Q_i), v_j= log(R_j), z_i⁽¹⁾= Q_i⁽¹⁾exp(Q_i)= Q_i⁽¹⁾z_i,
R_jv_j⁽¹⁾= R_j⁽¹⁾, $i=1,\ldots,h$, $j=1,\ldots,k$, $m\in {\bf N}_{0}$ },

where P^(m) is the derivative of order m of P for all $m \geq 1$.

EXAMPLE 2   Let

P=P(y,y⁽¹⁾,y⁽²⁾,exp(Q₁),exp(Q₂),log(R₁)= yy⁽²⁾-exp(y+(y⁽¹⁾)²)log(y+y⁽³⁾)y⁽¹⁾-(exp(y-y⁽²⁾))²y² (y⁽¹⁾)³,

where Q₁= y+(y⁽¹⁾)², Q₂=y-y⁽²⁾and R₁=y+y⁽³⁾.
If z₁= exp(Q₁), z₂= exp(Q₂) and v₁= log(R₁), then

P=yy⁽²⁾-z₁v₁y⁽¹⁾-z₂²y²(y⁽¹⁾)³.

The equation { P=0 } is equivalent to the system
$(\alpha)$={ P= yy⁽²⁾-z₁v₁y⁽¹⁾-z₂²y²(y⁽¹⁾)³=0, P⁽¹⁾=0,...,P^(m)=0, z₁= exp(y+(y⁽¹⁾)²), z₂= exp(y-y⁽²⁾))²), v₁= log(y+y⁽³⁾), z₁⁽¹⁾= (y⁽¹⁾+2y⁽¹⁾y⁽²⁾)z₁, z₂⁽¹⁾=
(y⁽¹⁾-y⁽³⁾)z₂, (y+y⁽³⁾)v₁⁽¹⁾= y⁽¹⁾+y⁽⁴⁾, $m\in {\bf N}_{0}$}.

Let $(\beta)$ be the following system of algebraic differential equations

$(\beta)$={ P= $P(y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots,v_{k})$=0, P⁽¹⁾=0,..., P^(m)=0, z_i⁽¹⁾= Q_i⁽¹⁾z_i, R_jv_j⁽¹⁾= R_j⁽¹⁾, $i=1,\ldots,h$, $j=1,\ldots,k$,
$m\in {\bf N}_{0}$ },

with $P \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k} \}$, $Q_{i}, R_{j} \in F\{ y \}$ for all i and j.
It is possible to find a system $(\gamma)$ of algebraic ordinary differential equations in the same differential variables $y,z_{1},\ldots,z_{h}, v_{1},\ldots,v_{k}$, that is equivalent to $(\beta)$, such that it contains a differential polynomial equation in the differential variable y. In other words we can find a different set of differential generators of the differential ideal I=[P, z_i⁽¹⁾-Q_i⁽¹⁾z_i, R_jv_j⁽¹⁾-R_j⁽¹⁾, $i=1,\ldots,h$, $j=1,\ldots,k$], that has the riquired properties.