Differential equation

A differential equation is an equation relating a function to its derivatives. It tells you how a quantity changes — the rate of change — rather than directly what the quantity is. Solving a differential equation means finding a function (or family of functions) whose derivatives satisfy the relationship.

The simplest example: $\frac{dy}{dx}=3x^2$. This tells you that the slope of $y$ at any point $x$ is $3x^2$. Integrating gives $y=x^3+C$ — a family of curves, one for each value of $C$. Each curve has the right slope at every point, but they’re shifted vertically by the constant.

To pin down a single specific curve, you need extra information — typically an initial condition like $y(0)=5$, which selects $C=5$ and gives $y=x^3+5$. That’s an Initial value problem.

Why differential equations matter

Most physical laws are stated as differential equations. Newton’s second law, $F=ma$, is really

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

— a second-order DE relating position $x$ to the force acting on the mass. Maxwell’s equations are partial differential equations describing electromagnetic fields. Heat flow, fluid dynamics, population growth, RLC circuits, beam deflection, chemical kinetics — all DE-driven.

The reason is that rates are easier to specify than absolute values. We rarely know “the position of every particle for all time”; we know “the force at any position,” from which we derive the position via the DE.

Two kinds

• Ordinary differential equations (ODEs) involve derivatives with respect to a single independent variable. Example: $\frac{dy}{dx}+2y=e^x$.
• Partial differential equations (PDEs) involve partial derivatives with respect to multiple variables. Example: $\frac{\partial u}{\partial t}=\alpha \frac{{\partial }^{2}u}{\partial x^2}$ (heat equation).

This vault focuses on ODEs.

Solutions

A function $\varphi (x)$ is a solution of a DE on an interval if its derivatives satisfy the equation everywhere on that interval. Different forms of solution:

• General solution: contains arbitrary constants — represents a family.
• Particular solution: a specific member of the family with constants pinned down by side conditions.
• Explicit solution: $y$ written as a function of $x$.
• Implicit solution: relates $x$ and $y$ but doesn’t isolate $y$.

For systematic methods to solve common types of DEs, see Separable equation, Integrating factor, Exact equation, Method of undetermined coefficients, Method of variation of parameters, Laplace transform.

Mathematical modeling with DEs

To build a mathematical model of a physical phenomenon:

1. Choose dependent and independent variables, set a frame of reference.
2. Choose units.
3. Identify the underlying principle (Newton’s laws, conservation, rate of growth, etc.).
4. Express it in terms of your chosen variables.
5. Solve by integrating.
6. Apply side conditions to pin down constants.

Example: a falling object under gravity. Newton: $F=ma$, with $F=-mg$, gives $\frac{d^{2}h}{d{t}^2}=-g$. Integrate twice:

$v(t)=\frac{dh}{dt}=-gt+c_1,h(t)=-\frac{1}{2}g{t}^2+c_{1}t+c_2$

Initial conditions $h(0)=1$, $v(0)=0$ give $c_1=0$, $c_2=1$, so $h(t)=-\frac{1}{2}g{t}^2+1$.