DIFFERENTIAL EQUATIONS 05
Autonomous Differential Equations
An autonomous differential equation gives the rate of change from the current state without explicit dependence on time.
The main idea
The current value of \(y\) determines the current rate.
1. Equilibria
Equilibria satisfy \(f(y)=0\). At such a state, \(dy/dt=0\), so a solution starting there remains there.
2. Direction of movement
If \(f(y)\gt0\), then \(y\) increases. If \(f(y)\lt0\), then \(y\) decreases.
3. Logistic example
For
\[y'=y(1-y),\]the equilibria are \(y=0\) and \(y=1\). For \(y\lt0\), the derivative is negative; for \(0\lt y\lt1\), it is positive; for \(y\gt1\), it is negative.
4. Stability
The sign pattern shows that \(y=1\) attracts nearby solutions, while \(y=0\) repels them.
5. Derivative test
If \(f'(y^*)\lt0\), an equilibrium is typically locally asymptotically stable. If \(f'(y^*)\gt0\), it is unstable.
6. Logistic population model
\[\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right),\qquad r\gt0,\quad K\gt0.\]For \(0\lt P\lt K\), population increases; for \(P\gt K\), it decreases. Thus \(K\) is a stable equilibrium.
7. Separation warning
Before dividing by \(f(y)\), first identify values where \(f(y)=0\), because these are equilibrium solutions and division by zero would remove them.