Ordinary Differential Equations 2 | Definitions

Ordinary differential equations (ODEs) are defined by how a function’s derivatives relate to the function itself—often through a rule that sets a combination of derivatives equal to zero. In this framework, an ODE of order K is any relation involving derivatives of a function up to the K-th derivative, with the highest derivative present determining the order. The unknown is a function X(t) defined on an interval I, and the equation uses a continuous function F that takes the independent variable t, the function value X(t), and the derivatives X′(t), X″(t), …, X^(K)(t) as inputs. A typical form is F(t, X(t), X′(t), …, X^(K)(t)) = 0, which makes the “order” a direct read-off from the highest derivative appearing.

A second key step is narrowing to the most common practical form: explicit first-order ODEs. “Explicit” means the derivative of interest appears alone on one side of the equation, while everything else sits on the other side. For a first-order explicit ODE, the structure is ẋ = W(t, x), where W is a continuous function defined on a product of intervals (t in I and x in J). This setup becomes even more powerful when multiple variables interact, because coupled equations can be packaged as a system. For example, two first-order equations for variables x1 and x2 can be combined into a vector equation Ẋ = W(t, X), where X(t) = (x1(t), x2(t)) and W maps into R^n. In the system form, the “state” is the vector of unknowns, and the rule W tells how that state changes with time t.

The definitions then pivot to what it means to actually solve an ODE. A solution is a differentiable function α(t) defined on some sub-interval (t0, t1) of the original time interval I. For a system, α(t) must land in the domain U (often an open set in R^n) so that W(t, α(t)) is well-defined. The solution condition is pointwise: for every t in the interval, the derivative α′(t) must equal W(t, α(t)). This is the operational test—plug the candidate function into the equation and verify it holds for all t in the solution’s domain.

To make the definition concrete, the course uses a simple two-dimensional system: x1̇ = x2 and x2̇ = −x1. Here W(t, X) = (x2, −x1), which does not explicitly depend on t. A specific solution is α(t) = (sin t, cos t), since differentiating gives α′(t) = (cos t, −sin t), matching W(t, α(t)). Geometrically, the solution’s image in the plane traces a circle of radius 1, called the orbit of the solution. Scaling the solution by 1/2 produces another valid solution, with an orbit that is a circle of radius 1/2. The takeaway is immediate: systems of ODEs can admit many different solutions, raising the next natural questions—whether solutions exist through prescribed points and whether such solutions are unique.