First-Order Differential Equations — Mathematics Knowledge System

Overview

A first-order differential equation relates a function to its own first derivative: dy/dx = f(x,y). Rather than solving for a number, you're solving for an entire function whose rate of change matches a given rule at every point. These equations model anything that changes based on its current state — population growth, radioactive decay, cooling objects, and circuit currents.

Intuition

Instead of asking 'what is y at x=5?', a differential equation asks 'given how y changes at every point, trace out the whole path.' It's like being told the compass heading at every location on a map, then asked to draw the trail. The initial condition (where you start) plus the rule for how the direction changes at each point (dy/dx = f(x,y)) uniquely determines the entire path forward.

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Formal Definition

Definition

The general first-order ODE and the special separable case:

\frac{dy}{dx} = f(x, y)

General form

\frac{dy}{dx} = g(x)h(y) \implies \int \frac{dy}{h(y)} = \int g(x)\,dx

Separable case

Notation

Notation	Meaning
$y' = f(x,y)$	First-order ODE, Lagrange notation
$y(x_0) = y_0$	Initial value problem — an ODE paired with a starting condition

Derivation

Solving the exponential growth equation dy/dx = ky by separation of variables:

\frac{dy}{y} = k\,dx

Separate variables — all y on one side, all x on the other

\int \frac{dy}{y} = \int k\,dx \implies \ln|y| = kx + C

Integrate both sides

y = Ae^{kx}

Exponentiate, absorbing the constant into A = e^C

Theorems

Theorem 1: Existence and Uniqueness (Picard–Lindelöf)

\text{If } f \text{ and } \partial f/\partial y \text{ are continuous near } (x_0, y_0), \text{ the IVP has a unique solution near } x_0.

Applications

Newton's law of cooling, radioactive decay, and simple harmonic motion are all modeled as first- or second-order ODEs.

Worked Examples

Separable equation, same form as exponential growth/decay.
$y = Ae^{-2x}$
Apply the initial condition y(0)=5 to find A.
$5 = Ae^0 \Rightarrow A = 5$

Answer: y = 5e^(-2x)

Practice Problems

Difficulty 6/10

Solve dy/dx = 3x² with y(0) = 1.

Difficulty 6/10

Freshly-poured concrete at 60°C cools in 20°C air following Newton's law of cooling dT/dt = −k(T − 20). If k = 0.05 per hour, what is the concrete's temperature after 10 hours?

Difficulty 6/10

A water tank drains so that its depth h obeys dh/dt = −0.2·h (a linearized outflow model), with initial depth 5 m. How deep is the water after 8 minutes, and what is the 'time constant' of the tank?

Difficulty 5/10

Radioactive decay follows dN/dt = −λN. Carbon-14 has a half-life of 5730 years. A timber pile from an old foundation retains 70% of its original C-14. Estimate its age.

Common Mistakes

Common Mistake

Forgetting the constant of integration when solving, or forgetting to apply the initial condition to pin it down.

A differential equation's general solution is a whole family of curves (one per value of C) -- only the initial condition selects the ONE curve that answers the actual problem.

Quiz

Flashcards

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Historical Background

Differential equations arose immediately alongside calculus itself — Newton's second law, F=ma, is literally a differential equation relating position to its second derivative. Leibniz and the Bernoulli brothers (Jacob and Johann) developed systematic solution methods (separation of variables, integrating factors) through the late 1600s and early 1700s while solving mechanics problems.

1690s
Jacob and Johann Bernoulli develop early solution techniques
Jacob Bernoulli, Johann Bernoulli
1694
Leibniz publishes on separable equations
Gottfried Wilhelm Leibniz

Summary

A first-order ODE relates y to its own derivative: dy/dx = f(x,y).
Separable equations can be solved by isolating x and y terms and integrating both sides.
An initial condition y(x₀)=y₀ selects one specific solution out of the whole family.
Picard-Lindelöf guarantees existence and uniqueness under mild continuity conditions.
Models exponential growth/decay, cooling, circuits, and population dynamics.

References

WebsiteWikipedia — Ordinary differential equation