First-order models · Definition

Equilibrium points

differential equationsapplications

Before you start you need to know

Before solving an equation it helps to know whether there are values at which the system simply stops changing.

Definition

An equilibrium point of an autonomous equation \(y'=f(y)\) is a constant value \(y^*\) of the unknown at which \(f\) vanishes.

\[f(y^*)=0\;\Longleftrightarrow\; y(t)\equiv y^*\text{ is a solution}\]

An equation is autonomous when its right-hand side depends only on \(y\), not on time. Then the rate of change is set by the current state, and the values where \(f(y)=0\) are resting states: if the system starts there, it stays there forever. They are also called constant solutions or critical points.

Finding them requires no integration, only solving the algebraic equation \(f(y)=0\). For instance, in \(y'=y^2-4y+3=(y-1)(y-3)\) the equilibria are \(y=1\) and \(y=3\). A repeated root counts as a single equilibrium, and an equation such as \(y'=y^2+1\) may have none.

By the existence and uniqueness theorem, distinct solutions never cross, so the other solution curves are trapped between consecutive equilibria and cannot pass through them. This gives the qualitative picture of the equation: between two consecutive equilibria the solution is monotone, because \(f\) keeps its sign.

The most common mistake is looking for equilibria in equations that are not autonomous: in \(y'=y-t\) the right-hand side changes with \(t\) and no constant makes it vanish at all times. Whether an equilibrium attracts or repels nearby solutions is a different question, stability, studied later and which, together with the logistic model, builds on this concept.

Think about it

Why does \(y'=y-t\) have no constant solutions?

If \(y\equiv c\), then \(y'=0\) but \(c-t\) is not zero for all \(t\).

Exercises 10

  1. Find the equilibrium point of \(y'=3y-6\).
  2. What are the equilibria of \(y'=y^2-4y+3\)?
  3. How many equilibria does \(y'=y(y-2)(y+1)\) have?
  4. Which of these equations is autonomous?
  5. How many real equilibria does \(y'=y^2+1\) have?
  6. How many equilibria does \(y'=\sin y\) have in the interval \([0,2\pi]\)?
  7. Find the equilibria of \(y'=2y\left(1-\tfrac{y}{10}\right)\).
  8. A tank satisfies \(y'=12-4y\). At what value does it stay constant?
  9. If \(y^*\) is an equilibrium and \(y(0)=y^*\), what is the solution of the initial value problem?
  10. How many distinct equilibria does \(y'=(y-1)^2(y+2)\) have?

References

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 12th edition, Wiley, 2021. §2.5
  • Dennis G. Zill, A First Course in Differential Equations with Modeling Applications, 11th edition, Cengage, 2018. §2.1
  • Steven H. Strogatz, Nonlinear Dynamics and Chaos, 2nd edition, Westview Press, 2015. §2.2