Autonomous System (mathematics) - Example

Example

The equation is autonomous, since the independent variable, let us call it, 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

Ffun = @(X,Y)(2-Y).*Y; % function f(x,y)=(2-y)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); % plot the direction field hold on; contour(X,Y,DY,); %add the isoclines title('Slope field and isoclines for f(x,y)=(2-y)y')

One can observe from the plot that the function is, of course, -invariant, and so it the shape of the solution, i.e. for any shift .

Solving the equation symbolically in MATLAB, by running

y=dsolve('Dy=(2-y)*y','x'); % solve the equation symbolically

we obtain two equilibrium solutions, and, and a third solution involving an unknown constant ,

y(3)=-2/(exp(C3 - 2*x) - 1)

Picking up some specific values for the initial condition, we can add the plot of several solutions

y1=dsolve('Dy=(2-y)*y','y(1)=1','x'); % solve the initial value problem symbolically y2=dsolve('Dy=(2-y)*y','y(2)=1','x'); % for different initial conditions y3=dsolve('Dy=(2-y)*y','y(3)=1','x'); y4=dsolve('Dy=(2-y)*y','y(1)=3','x'); y5=dsolve('Dy=(2-y)*y','y(2)=3','x'); y6=dsolve('Dy=(2-y)*y','y(3)=3','x'); ezplot(y1, ); ezplot(y2, ); % plot the solutions ezplot(y3, ); ezplot(y4, ); ezplot(y5, ); ezplot(y6, ); title('Slope field, isoclines and solutions for f(x,y)=(2-y)y')