Ordinary differential equation

An ordinary differential equation (ODE) is an equation involving an unknown function of one independent variable and its derivatives. Instead of solving for a number, we solve for a function.

Example:

$$\frac{dy}{dt}=ky.$$

Its solutions are

$$y(t)=C{e}^{kt},$$

which connects directly to Growth and decay.

Order and linearity

The order is the highest derivative appearing. Newtonian mechanics often gives second-order ODEs, because Newton’s Laws of Motion relates force to acceleration:

$$F=m\frac{d^{2}x}{d{t}^2}.$$

An ODE is linear if the unknown function and its derivatives only appear to the first power and are not multiplied together:

$$a_2(t)y^{\prime \prime }+a_1(t)y^{\prime }+a_0(t)y=g(t).$$

Linearity matters because solutions can be added and scaled when $g(t)=0$.

Initial and boundary conditions

A differential equation usually defines a family of functions. Extra data select one solution: initial conditions specify values at one point, while boundary conditions specify values at different points.

Common pitfalls

• Treating $dy/dt=ky$ like ordinary algebra without tracking the differential meaning.
• Forgetting constants of integration.
• Solving the ODE but not applying the initial or boundary conditions.

ODEs are the bridge from single-variable calculus to dynamics, modelling, and many areas of mathematical physics.