Ordinary Differential Equations 2 | Definitions [dark version]

TL;DR

An ODE’s order is determined by the highest derivative that appears in its defining relation.

Briefing Cornell Notes

Briefing

Ordinary differential equations are defined by how an unknown function’s derivatives relate to the independent variable and the function itself—typically through a continuous rule that sets a combination of derivatives equal to zero. In this setup, the “order” of the ODE is determined by the highest derivative that appears. That single structural idea matters because it standardizes how different differential problems are classified before any solving begins.

The foundation starts with function spaces: functions are taken from C^k, meaning they have derivatives up to order k and those derivatives are continuous. The independent variable is usually written as t, avoiding confusion with derivative notation (dots are used for derivatives like ẋ and ẍ). An ODE of order k is then expressed using a continuous function f that takes inputs (t, x(t), ẋ(t), …, x^(k)(t)) and forces the result to be zero. A concrete second-order example is given in the form t + x + 2ẋ + ẍ^2 = 0, illustrating how the highest derivative present (here ẍ) determines the order.

After laying out the general definition, the course narrows to a widely used special case: 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. This leads to a rule of the form ẋ = w(t, x), where w is a function of two inputs: the independent variable t and the dependent variable x. The transcript emphasizes that this common form is not a major restriction; it will later be shown to cover much of what matters.

The discussion then generalizes from a single equation to a system. When multiple dependent variables interact—such as ẋ1 depending on x2—those equations are treated together as a system rather than as separate independent ODEs. In vector form, an n-dimensional first-order system is written as Ẋ(t) = W(t, X(t)), where X(t) is an n-vector and W maps into R^n. The domain for the state variable is an open set U ⊂ R^n, reflecting that solutions must stay within the region where W is defined.

A solution is defined as a differentiable function α defined on some sub-interval (t0, t1) ⊂ I, mapping into U, that satisfies the system at every point in its domain: α̇(t) = W(t, α(t)). This pointwise condition is what turns the abstract rule into a concrete trajectory in state space.

To make the definition tangible, a simple two-dimensional system is used: ẋ1 = x2 and ẋ2 = −x1. Here W(t, x) = (x2, −x1) and W does not explicitly depend on t. The functions α(t) = (sin t, cos t) satisfy the system, producing an orbit that traces a circle of radius 1 in R^2. Scaling the solution by 1/2 yields another valid solution with an orbit that is a circle of radius 1/2, demonstrating that multiple solutions can exist for the same differential rule. That multiplicity sets up the next key question for the course: which solutions pass through specified points, and whether such solutions are unique.

Cornell Notes

The transcript builds the core definitions needed to study ordinary differential equations (ODEs): function regularity (C^k), the meaning of ODE order, explicit first-order ODEs, and systems of ODEs in vector form. An ODE of order k is determined by the highest derivative appearing in a relation f(t, x(t), ẋ(t), …, x^(k)(t)) = 0. For explicit first-order systems, the standard form is Ẋ(t) = W(t, X(t)), where X(t) is an n-vector and W maps into R^n. A solution is a differentiable function α(t) that stays in the state domain U and satisfies α̇(t) = W(t, α(t)) for every t in its interval. A worked example produces circular orbits, showing multiple solutions can exist.

How does the definition determine the “order” of an ordinary differential equation?

The order equals the highest derivative that appears in the defining relation. If the equation uses x^(k)(t) (the k-th derivative) as the top derivative term, then the ODE is of order k. In the example t + x + 2ẋ + ẍ^2 = 0, the highest derivative present is ẍ (second derivative), so it is a second-order ODE.

What does “C^k” mean for the functions used in ODE definitions?

C^k means the function has derivatives up to order k and those derivatives are continuous. So if x ∈ C^k on an interval I, then x^(j)(t) exists and is continuous for j = 0, 1, …, k. This regularity is what makes expressions involving derivatives well-defined and continuous.

What makes an ODE “explicit” in the course’s terminology?

An explicit first-order ODE places the derivative of interest by itself on one side. The transcript uses the form ẋ = w(t, x), where the right-hand side contains only t and x (no derivatives). This separation is what distinguishes explicit equations from more general implicit forms.

Why are coupled equations treated as a “system” rather than separate ODEs?

When one variable’s derivative depends on another variable’s value—like ẋ1 = x2 + t—those equations must be solved together because the variables interact. The transcript packages them into a vector equation Ẋ(t) = W(t, X(t)), where X(t) collects all dependent variables into one n-dimensional state.

What exactly qualifies a function α(t) as a solution to a system of ODEs?

A solution α is differentiable on some interval (t0, t1) and maps into the state domain U so that W(t, α(t)) is defined. It must satisfy the system pointwise: α̇(t) = W(t, α(t)) for every t in the interval. The derivative condition is the defining test.

How does the example system produce circular “orbits,” and what does that imply about solution multiplicity?

For ẋ1 = x2 and ẋ2 = −x1, the solution α(t) = (sin t, cos t) satisfies the system because d/dt(sin t) = cos t and d/dt(cos t) = −sin t. The trajectory (sin t, cos t) stays on the circle x1^2 + x2^2 = 1, giving radius 1. Scaling by 1/2 yields another valid solution with radius 1/2, showing the same differential rule can admit multiple solutions.

Review Questions

In an ODE defined by f(t, x(t), ẋ(t), …, x^(k)(t)) = 0, what feature of the expression fixes the order k?
For an explicit first-order system written as Ẋ(t) = W(t, X(t)), what two conditions must a candidate solution α(t) satisfy?
In the example ẋ1 = x2, ẋ2 = −x1, why does α(t) = (sin t, cos t) satisfy the system?

Key Points

1
An ODE’s order is determined by the highest derivative that appears in its defining relation.
2
C^k regularity ensures derivatives up to order k exist and are continuous, making ODE expressions well-defined.
3
Explicit first-order ODEs take the form ẋ = w(t, x), with the derivative isolated on one side.
4
Coupled equations involving multiple dependent variables are treated as a system and written compactly as Ẋ(t) = W(t, X(t)).
5
A solution is a differentiable function α(t) that stays in the state domain U and satisfies α̇(t) = W(t, α(t)) for all t in its interval.
6
The two-dimensional example ẋ1 = x2 and ẋ2 = −x1 generates circular orbits, illustrating that multiple solutions can exist for the same ODE system.
7
The next natural goal is to determine when a solution passing through specified points exists and whether it is unique.

Highlights

An ODE’s order is set purely by the highest derivative appearing in the equation, not by the complexity of the rest of the terms.

Explicit first-order ODEs are standardized as ẋ = w(t, x), which makes systems easier to express and analyze.

Writing coupled equations as a vector system Ẋ(t) = W(t, X(t)) captures variable interactions that separate scalar ODEs would miss.

In the example system ẋ1 = x2, ẋ2 = −x1, solutions trace circles: (sin t, cos t) gives radius 1, while scaling by 1/2 gives radius 1/2.

Topics

Ordinary Differential Equations
ODE Order
Explicit ODEs
Systems of ODEs
Solution Definition