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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:1. Ordinary Differential Equations
Subcategory:1.6. Systems of Two Nonlinear Equations
Equation(s):$\displaystyle x'_t={\lambda_1}x+{\lambda}x^2+{\gamma}xy+{\delta}x^2y+\frac{\lambda_1\gamma\delta}{{\lambda_2}{\lambda}}xy^2$,\hfill\break
$\displaystyle y'_t={\lambda_2}y+{\lambda}xy+{\gamma}y^2-\frac{\lambda_2\lambda\delta}{\lambda_1\gamma}}x^2y-{\delta}xy^2$,\hfill\break
where $\lambda_1\ne\lambda_2$,$\lambda$,$\gamma$,$\delta$ real-valued parameters with obvious limitations on their values.
Solution(s),
Transformation(s),
Integral(s)
:
General integral:\hfill\break
$\displaystyle x^{\lambda_2}({\lambda_1}{\lambda_2}+{\lambda_2}{\lambda}x+{\lambda_1}{\gamma}y)^\frac{(\lambda_1-\lambda_2)\lambda\gamma+\lambda_1\lambda_2\delta}{\lambda\gamma}=Cy^{\lambda_1}\exp\frac{\delta({\lambda_2}{\lambda}x+{\lambda_1}{\gamma}y)}{\lambda\gamma}$,\hfill\break
where $C$ is an arbitrary constant.
Remarks:Function \hfill\break
$\displaystyle  \mu(x,y)=\frac{1}{{\lambda_1}{\lambda_2}xy+{\lambda_2}{\lambda}x^2y+{\lambda_1}{\gamma}xy^2}$ is an integrating multiplier for the system.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Naziev
First name:Elbert
Country:Ukraine
City:Kiev
Affiliation:System of ODE
Statistic information
Submission date:Sat 03 Dec 2011 23:46
Edits by author:0

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