First-order ODEs

What is an ordinary differential equation?

A differential equation is an equation involving an unknown function and its derivatives. An ordinary differential equation, or ODE, involves derivatives with respect to one independent variable, often time.

A first-order ODE contains a first derivative but no higher derivatives. For example,

rac{dy}{dt} = ky

is a first-order ODE. It says the rate of change of $y$ is proportional to $y$ itself.

Differential equations are central in physics because laws often describe rates of change rather than direct formulas.

Growth and decay

The equation

rac{dy}{dt}=ky

models exponential growth if $k>0$ and exponential decay if $k<0$. The solution is

$y(t)=y_0e^{kt}$

where $y_0$ is the initial value.

For radioactive decay, the equation is often written

rac{dN}{dt}=-lambda N

with solution

$N(t)=N_0e^{-lambda t}$

This says the number of undecayed nuclei decreases at a rate proportional to the number remaining.

Separation of variables

Many first-order ODEs can be solved by separation of variables. The idea is to put all terms involving $y$ on one side and all terms involving $t$ on the other.

Starting with

rac{dy}{dt}=ky

rewrite as

rac{1}{y}dy = k,dt

Integrate both sides:

int rac{1}{y},dy = int k,dt

So

$ln|y| = kt + C$

Exponentiating gives

$y = Ae^{kt}$

where $A$ is determined by the initial condition.

Cooling and approach to equilibrium

Not all first-order equations grow or decay toward zero. Newton's law of cooling can be written

rac{dT}{dt}=-k(T-T_{env})

Here $T$ is object temperature and $T_{env}$ is environmental temperature. The temperature difference decays exponentially:

$T(t)-T_{env} = (T_0-T_{env})e^{-kt}$

This describes approach to equilibrium.

RC circuits

A charging or discharging capacitor in an RC circuit also follows first-order ODEs. During discharge,

rac{dQ}{dt}=-rac{1}{RC}Q

with solution

$Q(t)=Q_0e^{-t/RC}$

The quantity $RC$ is a time constant. It sets the timescale for decay.

Initial conditions

A differential equation usually has a family of solutions. An initial condition selects the physical solution. If you know $y(0)=y_0$, that determines the constant. In physics, initial conditions represent starting position, initial velocity, initial temperature, initial charge, or initial amount.

Direction fields

Even before solving an ODE exactly, you can interpret it. The derivative tells the slope of the solution at each point. A direction field sketches small slope marks showing how solutions flow. This helps visualize growth, decay, and equilibrium.

The big idea

First-order ODEs model systems where the rate of change depends on the current state. Exponential growth, exponential decay, cooling, drag models, and RC circuits all use this structure. Solving an ODE means finding a function whose behavior matches the rate law and initial condition.