Learning paths

Learning paths

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Full curriculumExplore every unit

Learning path

Full curriculum

Full curriculum
  1. Eigenvalues and eigenvectors
  2. Matrix multiplication
  3. Systems as mappings between signals
  4. Vector addition and scalar multiplication
  5. Introduction to ordinary differential equations
  6. Systems of first-order ordinary differential equations
  7. Deterministic linear state-space models
  8. Stochastic linear state-space models
  9. State-transition matrices and the matrix exponential
  10. Controllability of linear state-space systems
  11. Observability of linear state-space systems
  12. Linear ODE systems and eigenmodes

Unit content

Systems of first-order ordinary differential equations

Several coupled quantities can evolve together. A system of first-order ODEs can be written compactly as

$$\mathbf x'(t)=\mathbf f(t,\mathbf x).$$

The vector

$$\mathbf x=(x_1,\ldots,x_n)^T$$

collects the variables needed to describe the system's current state.

A higher-order scalar equation can be rewritten as a first-order system. For example, with

$$y''=F(t,y,y'),$$

define

$$x_1=y,\qquad x_2=y'.$$

Then

$$x_1'=x_2,\qquad x_2'=F(t,x_1,x_2).$$

This conversion is fundamental because first-order state equations provide a common representation for mechanics, circuits, control, numerical integration and dynamical systems.