Mixing
Solving a system of ODEs containing a product
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I’m a math student, but I’ve not yet taken a course on ODEs (though I’ve gone through a fair number of lecture notes and videos on the topic). In some of my personal work I ran into a system of ODEs roughly of the form
$$x_1’ = x_2x_3 + x_2$$
$$x_2’ = x_1x_3 + x_3$$
$$x_3’ = x_1x_2 + x_1$$
More generally, each first derivative is linear in products of the other variables. How could I go about solving the system above? What kinds of systems (if any) of this general form will admit analytics solutions?
ordinary-differential-equations
asked Jan 28 at 2:26
JFoxJFox
1747
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add a comment |
$begingroup$
I’m a math student, but I’ve not yet taken a course on ODEs (though I’ve gone through a fair number of lecture notes and videos on the topic). In some of my personal work I ran into a system of ODEs roughly of the form
$$x_1’ = x_2x_3 + x_2$$
$$x_2’ = x_1x_3 + x_3$$
$$x_3’ = x_1x_2 + x_1$$
More generally, each first derivative is linear in products of the other variables. How could I go about solving the system above? What kinds of systems (if any) of this general form will admit analytics solutions?
ordinary-differential-equations
asked Jan 28 at 2:26
JFoxJFox
1747
$endgroup$
$begingroup$
In general you need to be very lucky to find explicit solutions to nonlinear ODEs. For example, the system giving rise to the Lorenz attractor is about as simple-looking as your system. In your case there is a symmetry in the equations; maybe that can be used somehow?
$endgroup$
– Hans Lundmark
Jan 28 at 5:58
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You can try to find first integrals for this system of autonomous first-order ODEs, i. e. functions $I(x_1,x_2,x_3)$ which are constant on the solutions: $frac{d}{dt}I(x_1(t),x_2(t),x_3(t)) = 0$. For that purpose, the symmetric form of the system may be useful: begin{equation} frac{dt}{1} = frac{dx_1}{x_2(x_3+1)} = frac{dx_2}{(x_1+1)x_3} = frac{dx_3}{x_1(x_2+1)} end{equation} The knowledge of first integrals can help to reduce the number of unknowns.
$endgroup$
– Christoph
Jan 28 at 9:47
add a comment |
$begingroup$
I’m a math student, but I’ve not yet taken a course on ODEs (though I’ve gone through a fair number of lecture notes and videos on the topic). In some of my personal work I ran into a system of ODEs roughly of the form
$$x_1’ = x_2x_3 + x_2$$
$$x_2’ = x_1x_3 + x_3$$
$$x_3’ = x_1x_2 + x_1$$
More generally, each first derivative is linear in products of the other variables. How could I go about solving the system above? What kinds of systems (if any) of this general form will admit analytics solutions?
ordinary-differential-equations
asked Jan 28 at 2:26
JFoxJFox
1747
$endgroup$
I’m a math student, but I’ve not yet taken a course on ODEs (though I’ve gone through a fair number of lecture notes and videos on the topic). In some of my personal work I ran into a system of ODEs roughly of the form
$$x_1’ = x_2x_3 + x_2$$
$$x_2’ = x_1x_3 + x_3$$
$$x_3’ = x_1x_2 + x_1$$
More generally, each first derivative is linear in products of the other variables. How could I go about solving the system above? What kinds of systems (if any) of this general form will admit analytics solutions?
ordinary-differential-equations
ordinary-differential-equations
asked Jan 28 at 2:26
JFoxJFox
1747
asked Jan 28 at 2:26
JFoxJFox
1747
asked Jan 28 at 2:26
JFoxJFox
1747
asked Jan 28 at 2:26
JFoxJFox
1747
asked Jan 28 at 2:26
JFoxJFox
1747
1747
$begingroup$
In general you need to be very lucky to find explicit solutions to nonlinear ODEs. For example, the system giving rise to the Lorenz attractor is about as simple-looking as your system. In your case there is a symmetry in the equations; maybe that can be used somehow?
$endgroup$
– Hans Lundmark
Jan 28 at 5:58
$begingroup$
You can try to find first integrals for this system of autonomous first-order ODEs, i. e. functions $I(x_1,x_2,x_3)$ which are constant on the solutions: $frac{d}{dt}I(x_1(t),x_2(t),x_3(t)) = 0$. For that purpose, the symmetric form of the system may be useful: begin{equation} frac{dt}{1} = frac{dx_1}{x_2(x_3+1)} = frac{dx_2}{(x_1+1)x_3} = frac{dx_3}{x_1(x_2+1)} end{equation} The knowledge of first integrals can help to reduce the number of unknowns.
$endgroup$
– Christoph
Jan 28 at 9:47
add a comment |
$begingroup$
In general you need to be very lucky to find explicit solutions to nonlinear ODEs. For example, the system giving rise to the Lorenz attractor is about as simple-looking as your system. In your case there is a symmetry in the equations; maybe that can be used somehow?
$endgroup$
– Hans Lundmark
Jan 28 at 5:58
$begingroup$
You can try to find first integrals for this system of autonomous first-order ODEs, i. e. functions $I(x_1,x_2,x_3)$ which are constant on the solutions: $frac{d}{dt}I(x_1(t),x_2(t),x_3(t)) = 0$. For that purpose, the symmetric form of the system may be useful: begin{equation} frac{dt}{1} = frac{dx_1}{x_2(x_3+1)} = frac{dx_2}{(x_1+1)x_3} = frac{dx_3}{x_1(x_2+1)} end{equation} The knowledge of first integrals can help to reduce the number of unknowns.
$endgroup$
– Christoph
Jan 28 at 9:47
$begingroup$
In general you need to be very lucky to find explicit solutions to nonlinear ODEs. For example, the system giving rise to the Lorenz attractor is about as simple-looking as your system. In your case there is a symmetry in the equations; maybe that can be used somehow?
$endgroup$
– Hans Lundmark
Jan 28 at 5:58
$begingroup$
In general you need to be very lucky to find explicit solutions to nonlinear ODEs. For example, the system giving rise to the Lorenz attractor is about as simple-looking as your system. In your case there is a symmetry in the equations; maybe that can be used somehow?
$endgroup$
– Hans Lundmark
Jan 28 at 5:58
$begingroup$
You can try to find first integrals for this system of autonomous first-order ODEs, i. e. functions $I(x_1,x_2,x_3)$ which are constant on the solutions: $frac{d}{dt}I(x_1(t),x_2(t),x_3(t)) = 0$. For that purpose, the symmetric form of the system may be useful: begin{equation} frac{dt}{1} = frac{dx_1}{x_2(x_3+1)} = frac{dx_2}{(x_1+1)x_3} = frac{dx_3}{x_1(x_2+1)} end{equation} The knowledge of first integrals can help to reduce the number of unknowns.
$endgroup$
– Christoph
Jan 28 at 9:47
$begingroup$
You can try to find first integrals for this system of autonomous first-order ODEs, i. e. functions $I(x_1,x_2,x_3)$ which are constant on the solutions: $frac{d}{dt}I(x_1(t),x_2(t),x_3(t)) = 0$. For that purpose, the symmetric form of the system may be useful: begin{equation} frac{dt}{1} = frac{dx_1}{x_2(x_3+1)} = frac{dx_2}{(x_1+1)x_3} = frac{dx_3}{x_1(x_2+1)} end{equation} The knowledge of first integrals can help to reduce the number of unknowns.
$endgroup$
– Christoph
Jan 28 at 9:47
add a comment |
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In general you need to be very lucky to find explicit solutions to nonlinear ODEs. For example, the system giving rise to the Lorenz attractor is about as simple-looking as your system. In your case there is a symmetry in the equations; maybe that can be used somehow?
$endgroup$
– Hans Lundmark
Jan 28 at 5:58
$begingroup$
You can try to find first integrals for this system of autonomous first-order ODEs, i. e. functions $I(x_1,x_2,x_3)$ which are constant on the solutions: $frac{d}{dt}I(x_1(t),x_2(t),x_3(t)) = 0$. For that purpose, the symmetric form of the system may be useful: begin{equation} frac{dt}{1} = frac{dx_1}{x_2(x_3+1)} = frac{dx_2}{(x_1+1)x_3} = frac{dx_3}{x_1(x_2+1)} end{equation} The knowledge of first integrals can help to reduce the number of unknowns.
$endgroup$
– Christoph
Jan 28 at 9:47