Differential equation

Differential equations (or DEs, for short) are equations involving an unknown function $y(t)$ and one or more of its derivatives; it describes the behaviour of the function using its rate of change.

Oftentimes it’s easier to describe behaviour of systems by how they change in time and space instead of their present behaviour. So as a result DEs are commonly used as mathematical models of phenomena, because of how useful they are to predict and control the future behaviour of systems and events.

Terminology

There are a few key terminology we use to describe DEs:

• Ordinary: only involving one independent variable (usually $t$).
• Partial: where the unknown function has more than one variable.
• $n$-th order: the highest derivative in the equation is the $n$-th derivative.
• Linear: $y$ and its derivatives should be of degree one, i.e., no $\cos y$ or $y^2$.
• Autonomous: if the ODE doesn’t include $t$ as a parameter, and is only in terms of $y$, i.e., the independent variable doesn’t explicitly appear.

If we can express the ODE only in terms of $y$ one one side of the equation (i.e., where one side is 0), then we discuss a homogeneous ODE. Conversely if we have some function on the other side $g(t)$, then we have a non-homogeneous ODE.

Solutions to DEs are functions $y(t)$ defined over some interval $I\in \mathbb{R}$ that satisfy the equation with it and its derivatives. Determining these solutions can be difficult, so we have a few methods, the easiest of which are for first-order, then increasing in difficulty after that.

All $n$-th order ODEs have general solutions that depend on $n$ arbitrary constants, or parameters, such that:

$$y(t)=c_1{y}_1(t)+c_2{y}_2(t)+\cdots +c_n{y}_n(t)$$

Key concepts

Types

• First-order ODE
• Second-order ODE
• Partial differential equation
• Homogeneous ODE
• Non-homogeneous ODE

Terminology

• Solutions to differential equations
  • Reduction of order
  • Method of undetermined coefficients
  • Linear independence
    • Wronskian determinant
  • Laplace transform
• Numerical methods
  • Euler’s method
  • Runge-Kutta methods
  • Adams-Bashforth-Moulton method
• Initial-value problem
• Existence and uniqueness
• Equilibrium solution
• System of differential equations

Resources

• Ordinary Differential Equations: Analytical Methods and Applications, by Henner et al.
• Advanced Engineering Mathematics, by Dennis G. Zill

Courses

• MAT186 — Calculus I
• MAT187 — Calculus II
• MAT290 — Advanced Engineering Mathematics