What Is a Differential Equation?
What Is a Differential Equation?
A differential equation is an equation that involves an unknown function and one or more of its derivatives. While an ordinary algebraic equation asks "what number satisfies this relationship?", a differential equation asks "what function satisfies this relationship?" The answer is not a single number but an entire curve — or, more precisely, a family of curves determined up to one or more constants that are pinned down by initial conditions.
Differential equations are the language of change. Whenever a quantity's rate of change depends on its current value — population growth proportional to population size, a cooling cup of coffee losing heat in proportion to the temperature difference with its surroundings, a spring whose restoring force depends on displacement — the governing law is naturally expressed as a differential equation. Physics, biology, economics, and engineering are filled with such laws.
The order of a differential equation is the highest derivative that appears. A first-order equation involves only $y'$; a second-order equation involves $y''$; and so on. The order matters because it determines how many initial conditions you need to specify a unique solution. A first-order equation requires one condition (like $y(0) = 5$), while a second-order equation requires two (such as $y(0) = 1$ and $y'(0) = 0$).
A slope field is a visual tool that plots short line segments whose slopes match $y' = f(x,y)$ at a grid of points. Even before solving the equation analytically, the slope field lets you see the shape and behavior of solutions — where they increase, decrease, level off, or diverge. In this lesson you will learn to classify differential equations by order, interpret initial conditions, and read slope fields using the interactive explorer below.
$$y'=3x^2$$, $$\frac{dy}{dx}=ky$$, and $$y''+4y=0$$ are all differential equations.
The order is the highest derivative present. If $y''$ appears but no higher derivative does, the equation is second order.
An initial condition such as $y(0)=5$ selects one solution from a family.
Find the order of $y''+y'=0$.
- Second order.
Why is $y'=ky$ a growth model when $k>0$?
- The rate of change is proportional to the current amount
- so positive values tend to increase.
Practice Problems
Show Answer Key
1. Third order
2. An initial condition or specific function value
3. First order
4. The slope of solution curves at many points
5. Population growth, cooling, circuits, motion, and many others