Solution of the System of Two Coupled First-Order ODEs with Second-Degree Polynomial Right-Hand Sides

Abstract

The explicit solution $x_{n}\left (t\right ) ,$ n = 1,2, of the initial-values problem is exhibited of a subclass of the autonomous system of 2 coupled first-order ODEs with second-degree polynomial right-hand sides, hence featuring 12 a priori arbitrary (time-independent) coefficients:

$$ \dot{x}_{n}=c_{n1}\left( x_{1}\right)^{2}+c_{n2}x_{1}x_{2}+c_{n3}\left( x_{2}\right)^{2}+c_{n4}x_{1}+c_{n5}x_{2}+c_{n6}~,~~~n=1,2~. $$

The solution is explicitly provided if the 12 coefficients c_nj (n = 1,2; j = 1,2,3,4,5,6) are expressed by explicitly provided formulas in terms of 10 a priori arbitrary parameters; the inverse problem to express these 10 parameters in terms of the 12 coefficients c_nj is also explicitly solved, but it is found to imply—as it were, a posteriori—that the 12 coefficients c_nj must then satisfy 4 algebraic constraints, which are explicitly exhibited. Special subcases are also identified the general solutions of which are completely periodic with a period independent of the initial data (“isochrony”), or are characterized by additional restrictions on the coefficients c_nj which identify particularly interesting models.

Appendices

Appendix : A

In this Appendix A we tersely demonstrate the following elementary fact, which clearly implies the result (6): that the solution of the initial-values problem for the ODE

$$ \dot{y}\left( t\right) =a_{2}\left[ y\left( t\right) \right]^{2}+a_{1}y\left( t\right) +a_{0}~, $$

(74a)

is provided by the formula

$$ y\left( t\right) =\frac{y_{+}\left[ y\left( 0\right) -y_{-}\right] -y_{-} \left[ y\left( 0\right) -y_{+}\right] \exp \left( \beta t\right) }{y\left( 0\right) -y_{-}-\left[ y\left( 0\right) -y_{+}\right] \exp \left( \beta t\right) }~, $$

(74b)

with y_± defined as follows:

$$ y_{\pm }=\left( -a_{1}\pm \beta \right) /\left( 2a_{2}\right) ~,~~~\beta = \sqrt{\left( a_{1}\right)^{2}-4a_{0}a_{2}}~. $$

(74c)

Indeed the ODE (74a) can clearly be reformulated as follows:

$$ \dot{y}=a_{2}\left( y-y_{+}\right) \left( y-y_{-}\right) $$

(75a)

with y_± defined by (74c); and then (again, via (74c)) this ODE can be rewritten as follows:

$$ \dot{y}\left[ \left( y-y_{+}\right)^{-1}-\left( y-y_{-}\right)^{-1}\right] =\beta ~. $$

(75b)

The integration of this ODE for the dependent variable $y\left (t^{\prime }\right )$ over the independent variable $t^{\prime }$—from $t^{\prime }=0$ to $t^{\prime }=t$—clearly yields

$$ \ln \left[ \frac{y\left( t\right) -y_{+}}{y\left( 0\right) -y_{+}}\right] -\ln \left[ \frac{y\left( t\right) -y_{-}}{y\left( 0\right) -y_{-}}\right] =\beta t~, $$

(76)

which coincides—after exponentiation—with (74b). Q. E. D.

Finally let us emphasize that the results reported above are valid for generic values of the parameters; the interested reader shall have no difficulty to figure out the results in special cases, for instance those with a₂ = 0 or β = 0.

Appendix : B

In this Appendix ?? we tersely outline the derivation of the expressions (??) and (??) of the 12 parameters c_nj in terms of the 10 parameters A_nm and a_nℓ.

The first step is to invert the relations (??), getting

$$ y_{1}\left( t\right) =\left[ A_{22}x_{1}\left( t\right) -A_{12}x_{2}\left( t\right) \right] /D~, $$

(77a)

$$ y_{2}\left( t\right) =\left[ A_{11}x_{2}\left( t\right) -A_{21}x_{1}\left( t\right) \right] /D~, $$

(77b)

where the quantity D is defined as above, see (??).

The second step is to note that the relations (??) imply

$$ \dot{x}_{n}=A_{n1}\dot{y}_{1}+A_{n2}\dot{y}_{2}~,~~~n=1,2~, $$

(78a)

hence, via the ODEs (??),

$$ \begin{array}{@{}rcl@{}} \dot{x}_{n}=A_{n1}\left[ a_{12}\left( y_{1}\right)^{2}+a_{11}y_{1}+a_{10} \right] && \\ +A_{n2}\left[ a_{22}\left( y_{2}\right)^{2}+a_{21}y_{2}+a_{20}\right] ~,~~~n=1,2~. && \end{array} $$

(78b)

The third and last step is to insert the expressions (??) of y₁ and y₂ in terms of x₁ and x₂ in the right-hand sides of these ODEs. Then, via a bit of trivial if tedious algebra, there obtains the system (??) with the definitions (??) and (??) of the 12 coefficients c_nj. Q. E. D.

Appendix : C

In this Appendix ?? we show that the solutions $x_{n}\left (t\right )$ of the dynamical system (??)—as treated above, see Sections ?? and ??—do not depend on the free parameters λ_n introduced via the positions (??).

To this end we insert the expressions (??) of the parameters A_nm in terms of the free parameters λ_n in the formulas (??) and (??) expressing the 6 parameters a_nk; in order to display the very simple dependence of these 8 parameters from the 2 free parameters λ_n. We thus easily find the following formulas:

$$ a_{n\ell }\equiv \left( \lambda_{n}\right)^{\ell -1}\alpha_{n\ell }\left( C\right) ~,~~~n=1,2~,~~~\ell =0,1,2~, $$

(79a)

where the notation C indicates—above and hereafter—the set of the 12 parameters c_nj, and the 6 functions $\alpha _{n\ell }\left (C\right )$ are explicitly displayed in Section ??, see (??); of course to this end we also used the definitions (??) of the 2 auxiliary parameters z_n in terms of the 4 coefficients c_nm (n = 1, 2; m = 1, 2).

The next step is to insert the formulas (79a) in the expressions (??), getting thereby

$$ y_{n\pm }\equiv \left( \lambda_{n}\right)^{-1}w_{n\pm }\left( C\right) ~,~~~\beta_{n}\equiv \beta_{n}\left( C\right) ~,~~~n=1,2~, $$

(79b)

again with the functions $w_{n\pm }\left (C\right )$ and $\beta _{n}\left (C\right )$ explicitly displayed in Section ??, see (??) and (??).

The insertion of these formulas in the expressions (??) of the solutions $y_{n}\left (t\right )$ of the auxiliary dynamical system (??) evidences the following, very simple, dependence of these functions from the free parameters λ_n:

$$ y_{n}\left( t\right) \equiv \left( \lambda_{n}\right)^{-1}w_{n}\left( C,t\right) ~,~~~n=1,2~, $$

(80)

where again the 2 functions $w_{n}\left (C,t\right )$ are explicitly displayed in Section ??, see (??).

And via the insertion in the expressions (??) of $x_{n}\left (t\right )$ of these formulas (80), together with the expressions (??) of A_nm, we conclude that the solutions $x_{n}\left (t\right )$ are independent of the free parameters λ_n; as indeed displayed in Section 4, see the set of eqs. from (39) to (44). Q. E. D.