Introduction to Differential Equations
A DE is an equation relating an unknown function and its derivatives. Solutions are functions, not numbers.
IWhat is a Differential Equation?
A differential equation (DE) is an equation that contains an unknown function and one or more of its derivatives. Unlike algebra where we solve for a number, here we solve for a function.
The simplest possible DE:
This says "the rate of change of y with respect to x equals 2x". The solution is not a number — it is a whole family of curves:
Every value of C gives a different parabola. Together they form the general solution.
The word "differential" refers to the differentials and . A DE is simply any equation that links a function to how it changes.
IIGeneral vs Particular Solution
A general solution contains one or more arbitrary constants. It represents a whole family of curves.
A particular solution is obtained by giving the constants specific values — usually from an initial condition such as or .
The number of arbitrary constants equals the order of the DE.
An initial condition is a known value of the function (or its derivative) at a specific point. It "pins down" one member of the family of curves.