Systems of Nonalgebraic Ordinary Differential Equations

Let R be a differential ring of functions and let P_l= $P_{l}(y_{1},\ldots,y_{1}^{(n_{1l})},\ldots,y_{m},\\ \ldots,y_{m}^{(n_{ml})},z_{1},\ldots,z_{h},v_{1}, \ldots,v_{k})$ in $R\{ y_{1},\ldots,y_{m},z_{1},\ldots,z_{h},v_{1},\ldots, v_{k} \}$ with
$l=1,\ldots,p$

Let z_i= exp(Q_i) and let v_j= log(R_j), where Q_i= $Q_{i}(y_{1},\ldots,y_{1}^{(q_{i1})},\ldots, y_{m},\\ \ldots,y_{m}^{(q_{im})})$ and R_j= $R_{j}(y_{1},\ldots,y_{1}^{(r_{j1})},\ldots,y_{m},\ldots, y_{m}^{(r_{jm})})$ are polynomial functions with coefficients in the differential ring R for all $i=1,\ldots,h$ and all $j=1,\ldots,k$, e.g. $Q_{i}, R_{j} \in R\{y_{1},\ldots,y_{m}\}$ for all i and j. The system of ordinary differential equations

$(\alpha)$={ P_l= $P_{l}(y_{1},\ldots,y_{1}^{(n_{1l})},\ldots,y_{m}, \ldots,y_{m}^{(n_{ml})},exp(Q_{1}),\ldots,\\ exp(Q_{h}),log(R_{1}),\ldots, log(R_{k}))$=0, $l=1,\ldots,p$ }

is in general a system of nonlinear and nonalgebraic differential equations.

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 system $(\alpha)$is equivalent to the system

$(\beta)$={ P_l= $P_{l}(y_{1},\ldots,y_{1}^{(n_{1l})},\ldots,y_{m},\\ \ldots,y_{m}^{(n_{ml})},z_{1},\ldots,z_{h},v_{1}, \ldots,v_{k})$=0, P_l⁽¹⁾=0,..., P_l^(m_l)=0, z_i= exp(Q_i), v_j= log(R_j), z_i⁽¹⁾=
Q_i⁽¹⁾z_i, R_jv_j⁽¹⁾= R_j⁽¹⁾, $l=1,\ldots,p$ $i=1,\ldots,h$, $j=1,\ldots,k$,
$m_{l} \in {\bf N}_{0}$ },

where P^(m_l) is the derivative of order m_l of P_l for all $m_{l} \geq 1$.

Let N=max{n_tl,q_it,r_jt: $t=1,\ldots,m$, $l=1,\ldots,p$, $i=1,\ldots,h$, $j=1,\ldots,k$}. By using results from section 3, the system $(\alpha)$ is equivalent to the system

$(\alpha')$={ P_l= $P_{l}(y_{1},\ldots,y_{1}^{(n_{1l})},\ldots,y_{m},\\ \ldots,y_{m}^{(n_{ml})},z_{1},\ldots,z_{h},v_{1}, \ldots,v_{k})$=0, z_i⁽¹⁾= Q_i⁽¹⁾z_i, R_jv_j⁽¹⁾= R_j⁽¹⁾, P_1l1= $P_{1l_{1}}(y_{1},\ldots,y_{1}^{(N+1)},\ldots,y_{m},\\ \ldots,y_{m}^{(N+1)},z_{1},\ldots,z_{h}, v_{1},\ldots,v_{k-1})$=0,..., P_klk= $P_{kl_{k}}(y_{1},\ldots,\\ y_{1}^{(N+k)},\ldots,y_{m}, \ldots,y_{m}^{(N+k)},z_{1},\ldots,z_{h})$=0,..., P_(k+h)lk+h=
$P_{(k+h)l_{k+h}}(y_{1},\ldots,y_{1}^{(N+h+k)},\ldots,y_{m}, \ldots,y_{m}^{(N+k+h)})$=0, z_i= exp(Q_i), v_j= log(R_j), $l=1,\ldots,p$, $i=1,\ldots,h$, $j=1,\ldots,k$, $l_{r}=1,\ldots,\\ r(l)$, $r=1,\ldots,k+h$},

EXAMPLE 7   Let C(p,q)=qexp(p) be the conservative function and let D(p,q)=p+q be the dissipative function. Let's consider the corresponding nonconservative system

$(\alpha)$={ $p^{(1)}-\frac{\partial}{\partial q}C- \frac{\partial}{\partial p}D=p^{(1)}-exp(p)-1$=0, $q^{(1)}+\frac{\partial}{\partial p}C- \frac{\partial}{\partial q}D=q^{(1)}+qexp(p)-1$=0}.

Let z=exp(p). The system $(\alpha)$ is equivalent to the system

$(\beta)$={ p⁽¹⁾-z-1=0, q⁽¹⁾+qz-1=0, z-exp(p)=0, z⁽¹⁾-p⁽¹⁾z=0}.

By eliminating z the system is equivalent to tha system

$(\gamma)$={ p⁽¹⁾-z-1=0, q⁽¹⁾+qz-1=0, p⁽²⁾-(p⁽¹⁾)²+p⁽¹⁾=0, z-exp(p)=0, z⁽¹⁾-p⁽¹⁾z=0}. $(p(t)=log(\frac{exp(t)}{1-exp(t+c)})+c$,
$q(t)=(1-exp(t+c)) \int\frac{1}{(1-exp(t+c))^{2}}+(1-exp(t+c))c_{1}$),

is the solution of the system, where c, c₁ are arbitrary constants.