Definition

A differential equation of the form y′(t) = f(t, y(t)), y(t₀) = y₀ is said to be a first-order differential equation because only the first derivative of the dependent variable y(t) appears. If derivatives of order 2 appeared, the equation would be of second order, and so on.

Example

Consider the first-order differential equation:

y′(t) = t    ,    y(0) = 1

This is one of the simplest differential equations one can imagine. Integrating on both sides gives:

∫ y′(t) dt = ∫ t dt

which leads to:

y(t) = t²/2 + C

This is the general solution. The constant C is determined using the initial condition:

y(0) = 1 ⇒ C = 1

Therefore, the particular solution is:

y(t) = t²/2 + 1

Example

Consider the differential equation:

y′(t) = t·y(t)    ,    y(1) = 2

Direct integration is not possible here. We begin by separating variables:

dy / y = t dt

Integrating gives:

ln y = t²/2 + C

Hence, the general solution is:

y(t) = C · e^t²/2

Applying the initial condition:

y(1) = 2 = C·e^1/2 ⇒ C = 2e^−1/2

Thus, the particular solution is:

y(t) = 2e^(t²−1)/2

Classical references such as Simmons, Derrick & Grossman contain many similar examples, but our focus is on numerical methods for solving differential equations.

Numerical Perspective

Numerical techniques do not provide the exact solution for all values of t. Instead, they compute approximate values of the solution only at discrete points t_i, separated by a step size:

h = t_i+1 − t_i

In most methods, this step size h is constant. We denote the exact solution by y(t_i) and the numerical approximation by y_i.