01 / From numbers to functions
The unknown is a function
An ordinary equation asks which number works. A differential equation asks which whole curve follows a rule from start to finish. The answer you are looking for is a function.
When you see 2a + 1 = 7, you are looking for a number: a = 3. Put 3 back in, both sides match, and you are done.
Now try a different question. A quantity changes as x changes, and at every position, its rate of change equals its current value. Which function can keep following that rule?
That is a differential equation. It puts an unknown function and its derivative into one equation and states the relationship they must satisfy. What is unknown here is the whole of y(x), not a single number at one position.
A function gives you results
y(x) = eˣ
Give it an x, and you can compute y.
A differential equation gives you a rule
y′(x) = y(x)
You don’t know the whole function yet, but you know how its rate of change and its value are tied together.
The two run in opposite directions. Start from a function, and you can differentiate it to check which rule it follows. Start from a differential equation, and you look for the functions that follow the rule.
02 / Every symbol has a job
y is height, y′ is slope
y is how high the curve is right now. y′ is how fast it is changing right now. The equation on this page asks those two numbers to be exactly equal at every point.
x is the independent variable: you move along it to watch things change. y(x) is the dependent variable: its value depends on x. On the plot, x runs along the horizontal axis and y is the height.
y′(x) is read “y prime of x.” It is the derivative of y, also written dy/dx. It describes how y responds when x changes a little. On the graph, it is the slope of the tangent line at that point.
For example, if y′ = 2 at some point and x moves right by a small 0.01, y rises by about 0.02. A negative derivative means the curve falls as you move right. A derivative of 0 means the tangent line is horizontal.
Need a refresher? Where does the derivative come from?
Take two points a distance h apart. The average rate of change is [y(x+h)−y(x)]/h, the slope of the line through those two points.
Now let h get closer and closer to 0. If the ratio has a limit, that limit is the derivative at the point:
A limit does not mean setting h to 0. That would divide by zero and mean nothing. It asks where the ratio heads as h approaches 0.
So y′ = y means something concrete. Where the height is 2, the equation demands a slope of 2. Where the height is −1, it demands a slope of −1. Where the height is 0, the curve must be flat.
Slope is Δy/Δx, not height. This equation specifically requires the two numbers to be equal, and that is what ties height to slope. Most functions have no such property.
03 / Substitute, step by step
Why is eˣ a solution?
To check whether a function is a solution, find its derivative, put both back into the equation, and confirm that it holds at every point of the interval you chose.
The natural exponential function eˣ has a special property: differentiate it with respect to x and you get eˣ back. That is the calculus fact this example relies on. e ≈ 2.71828 is a fixed constant; only x varies.
Propose a candidate. This only names the function to check. Nothing is proved yet.
y(x) = eˣFind its derivative. Use the derivative rule for the natural exponential.
y′(x) = eˣSubstitute into both sides of the equation.
Left side: y′(x) = eˣRight side: y(x) = eˣCompare. The two sides agree for every real x.
y′(x) − y(x) = 0
So y(x)=eˣ satisfies the equation on the whole real line. It is a solution. That conclusion comes from the identity you get after substituting; dragging the slider only lets you see it.
“A solution” always comes with an interval
A solution must have the derivatives the equation needs on the interval you are considering, and it must satisfy the equation at every point there. If a formula is undefined at some point, you can’t pretend it is still one solution across that point.
Here eˣ is defined and differentiable for every real number, so there is no need to split the interval. Later, when denominators, logarithms or solutions that diverge appear, finding the interval becomes part of solving.
04 / An easy place to slip
Matching at one point is not a solution
Being right at one point doesn’t make a function a solution. The equation has to hold at every point along the way, and one point where it fails is enough to rule the function out.
Now look at the candidate y=x+1. It is a straight line with a constant slope of 1, so y′=1.
| x | Height y | Derivative y′ | Equal? |
|---|---|---|---|
| 0 | 1 | 1 | Equal at this point |
| 1 | 2 | 1 | Not equal |
At x = 0, the line happens to have the same height and the same tangent as eˣ. But as you move right, the line’s slope stays at 1, while the slope the equation demands grows with the height. The line matches at a single point, so it is not a solution of y′ = y on any interval of nonzero length.
Back in the plot, choose “y = x + 1,” stop at x = 0, then move right. The gold actual tangent and the blue required direction come apart.
Two constant functions that look equally simple
y=1 has derivative 0, so the equation would require 0=1, which is false. y=0 also has derivative 0, but now the equation requires 0=0, which holds everywhere.
The zero function is a complete solution too. A solution doesn’t have to rise, oscillate, or even curve.
05 / What the equation describes
Describing real change with an equation
One equation can describe many real situations. First, though, you have to say what each symbol stands for and what units it carries.
If a quantity changes at a rate proportional to its current size, you can start to model it with dy/dt=ky. Here k is the constant of proportionality. Positive, negative and zero values of k describe different rules of change.
When t measures time, k must have units of “per unit of time” so that both sides match. This page’s y′=y is a dimensionless math example, the case where that constant is 1. It does not claim that any real system grows at this rate.
Setting up a differential equation, solving it, and checking whether the model fits reality are three related but separate jobs. A perfectly correct mathematical solution can still rest on physical assumptions that don’t apply.
How do ordinary and partial differential equations differ?
When the unknown function has a single independent variable, such as y(x), an equation with ordinary derivatives is an ordinary differential equation, or ODE.
If temperature varies with both position and time, the function has more than one independent variable. An equation describing change in different directions may then contain partial derivatives. That is a partial differential equation, or PDE. This series starts with ODEs.
06 / Put the idea to work
Try three more examples
Decide for yourself first. Then open the reasoning and see where it differs from yours.
1. Is y = 3eˣ also a solution of y′ = y?
See the full reasoning
The constant 3 stays when you differentiate: y′=3eˣ. The right side, y, is also 3eˣ, so the equation holds for every real x. It is a solution as well. The equation does not single out the one curve eˣ.
2. Is y = eˣ + 2 still a solution of the same equation?
See the full reasoning
The derivative of the constant 2 is 0, so y′=eˣ. But y=eˣ+2. The two always differ by 2, so it is not a solution. Multiplying a function by a constant and adding a constant to it are different operations.
3. If you check many points on the plot and they all match, is that a proof?
See the full reasoning
A finite number of points is evidence only about those points; the interval holds other points too. To prove that this particular candidate is a solution, differentiate and substitute as above, and arrive at an identity that holds across the whole interval. Finding one point where the equation fails is enough to rule the function out on any interval that contains that point.
You have now found eˣ, 3eˣ and 0. Why does one rule of change leave so many solutions? “Solution Families and Initial Conditions” uses an initial condition to narrow a whole family of curves down to one.
Check and go further
Sources and scope
This page is about ordinary differential equations with one independent variable, and it uses dimensionless math examples. The plot shows only a finite window. A curve leaving the window does not mean the solution ends there.
The pictures help you see; substitution is what checks that an example satisfies the equation. The general theory of existence and uniqueness, and more ways to solve equations exactly, come in later topics.
- Jiří Lebl · Introduction to differential equations: what equations and solutions are, and how to check a solution.
- Jiří Lebl · Slope fields: slope fields and the conditions for solutions of initial value problems.
- Jiří Lebl · Euler’s method: step-by-step approximation and its error.
- MIT OpenCourseWare · First Order Differential Equations: conceptual, geometric and numerical views of first-order equations.