Editing Autonomous system (mathematics) (section)

==Example==
The equation <math>y'= \left(2-y\right)y</math> is autonomous, since the independent variable (<math>x</math>) does not explicitly appear in the equation. 
To plot the [[slope field]] and [[isocline]] for this equation, one can use the following code in [[GNU Octave]]/[[MATLAB]]
<syntaxhighlight lang="matlab">
Ffun = @(X, Y)(2 - Y) .* Y; % function f(x,y)=(2-y)y
[X, Y] = meshgrid(0:.2:6, -1:.2:3); % choose the plot sizes
DY = Ffun(X, Y); DX = ones(size(DY)); % generate the plot values
quiver(X, Y, DX, DY, 'k'); % plot the direction field in black
hold on;
contour(X, Y, DY, [0 1 2], 'g'); % add the isoclines(0 1 2) in green
title('Slope field and isoclines for f(x,y)=(2-y)y')
</syntaxhighlight>
One can observe from the plot that the function <math>\left(2-y\right)y</math> is <math>x</math>-invariant, and so is the shape of the solution, i.e. <math>y(x)=y(x-x_0)</math> for any shift <math>x_0</math>.

Solving the equation symbolically in [[MATLAB]], by running 
<syntaxhighlight lang="matlab">
syms y(x);
equation = (diff(y) == (2 - y) * y);
% solve the equation for a general solution symbolically
y_general = dsolve(equation);
</syntaxhighlight>
obtains two [[Equilibrium point|equilibrium]] solutions, <math>y=0</math> and <math>y=2</math>, and a third solution involving an unknown constant <math>C_3</math>,
<syntaxhighlight lang="matlab" inline>-2 / (exp(C3 - 2 * x) - 1)</syntaxhighlight>.

Picking up some specific values for the [[initial condition]], one can add the plot of several solutions
[[File:Slop field with isoclines and solutions.png|thumb|Slope field with isoclines and solutions]]
<syntaxhighlight lang="matlab">
% solve the initial value problem symbolically
% for different initial conditions
y1 = dsolve(equation, y(1) == 1); y2 = dsolve(equation, y(2) == 1);
y3 = dsolve(equation, y(3) == 1); y4 = dsolve(equation, y(1) == 3);
y5 = dsolve(equation, y(2) == 3); y6 = dsolve(equation, y(3) == 3);
% plot the solutions
ezplot(y1, [0 6]); ezplot(y2, [0 6]); ezplot(y3, [0 6]);
ezplot(y4, [0 6]); ezplot(y5, [0 6]); ezplot(y6, [0 6]);
title('Slope field, isoclines and solutions for f(x,y)=(2-y)y')
legend('Slope field', 'Isoclines', 'Solutions y_{1..6}');
text([1 2 3], [1 1 1], strcat('\leftarrow', {'y_1', 'y_2', 'y_3'}));
text([1 2 3], [3 3 3], strcat('\leftarrow', {'y_4', 'y_5', 'y_6'}));
grid on;
</syntaxhighlight>