Systems and stability · Definition

System of differential equations

differential equationssystems

Before you start you need to know

It gathers several quantities that change and influence each other, something a single equation cannot describe: competing populations, coupled circuits, chained springs.

Definition

A set of \(n\) equations that fix, through a vector function \(\mathbf F\), the derivatives of \(n\) unknown functions \(x_1(t),\dots,x_n(t)\) of a single variable \(t\).

\[\mathbf x'=\mathbf F(t,\mathbf x),\qquad \mathbf x=(x_1,\dots,x_n)\]

With one unknown a single equation suffices; when two or more quantities affect each other, each derivative may depend on all the others. For instance, in \(x'=-y,\ y'=x\) the rate of change of \(x\) is set by \(y\) and vice versa. If \(\mathbf F\) does not depend explicitly on \(t\), the system is called autonomous, and its solution can then be drawn as a curve in the space of the unknowns.

Every equation of order \(n\) reduces to a first-order system: take \(y,y',\dots,y^{(n-1)}\) as the unknowns and solve for the highest derivative. Thus \(y''+4y=0\) becomes \(x_1'=x_2,\ x_2'=-4x_1\). This reduction is why it is enough to study first-order systems.

A system together with the values of the \(n\) unknowns at an instant \(t_0\) is an initial value problem; under smoothness hypotheses on \(\mathbf F\) it has a unique solution. An equilibrium is a point where \(\mathbf F=\mathbf 0\), so the constant solution at that point solves the system.

When \(\mathbf F\) is linear in \(\mathbf x\), the system is written \(\mathbf x'=A(t)\mathbf x+\mathbf g(t)\), and linear-algebra tools apply. The constant-coefficient case, the next concept in the course, is solved with eigenvalues and eigenvectors. A common mistake is to forget an initial condition: a system of \(n\) equations needs \(n\) conditions.

Think about it

Turn \(y''+2y'+5y=0\) into a first-order system.

With \(x_1=y,\ x_2=y'\): \(x_1'=x_2,\ x_2'=-5x_1-2x_2\).

Exercises 10

  1. With \(x_1=y\) and \(x_2=y'\), the equation \(y''-3y'+2y=0\) is equivalent to \(\mathbf x'=A\mathbf x\). What is \(A\)?
  2. How many equations does the first-order system equivalent to a third-order equation have?
  3. The system \(x'=y,\ y'=4x\) has the solution:
  4. Find the equilibrium point of \(x'=x-y,\ y'=x+y-2\).
  5. Solve \(x'=2x,\ y'=-3y\) with \(x(0)=1,\ y(0)=2\).
  6. Turn \(y''+4y=0\) into a system with \(x_1=y,\ x_2=y'\). Which system is it?
  7. If \(x'=y\) and \(y'=-x\), what is \(x''\)?
  8. How many initial conditions determine a unique solution of a system of \(3\) first-order equations?
  9. Which of these systems is decoupled (each equation involves a single unknown)?
  10. How many equilibrium points does \(x'=x(1-y),\ y'=y(x-1)\) have?

References

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 12th edition, Wiley, 2021. §7.1
  • Dennis G. Zill, A First Course in Differential Equations with Modeling Applications, 11th edition, Cengage, 2018. §8.1
  • Steven H. Strogatz, Nonlinear Dynamics and Chaos, 2nd edition, Westview Press, 2015. §5.0