What a Differential Equation Is
Chapter Eighty
Syllabus topic Module 2, "2.1 Equation of the first order and of the first degree"
Pages 176 to 177 of 303
In one line
A differential equation is an equation that relates a quantity to its own rate of change, and solving it means finding the function, not a number.
The definition
A differential equation is an equation containing one or more derivatives of an unknown function.
Compare it with an ordinary equation.
| An algebraic equation | A differential equation | |
|---|---|---|
| Example | x squared minus 5x plus 6 = 0 | dy/dx = 2y |
| The unknown is | a number | a function |
| A solution is | a number that fits | a function that fits |
| How many solutions | finitely many | a whole family |
| Checking a solution | substitute the number | substitute the function and its derivatives |
The third row is the one to take seriously. Solving x squared minus 5x plus 6 = 0 gives two numbers. Solving dy/dx = 2y gives every function of the form C e to the 2x, one for each value of C, which is infinitely many.
Ordinary and partial
If the unknown function has one independent variable, its derivatives are ordinary derivatives and the equation is an ordinary differential equation. If it has more than one, the derivatives are partial and the equation is a partial differential equation.
This paper is entirely about ordinary differential equations. MU's label says "Ordinary Linear Differential Equations" explicitly.
What a solution is
A function is a solution of a differential equation on an interval if substituting it, and its derivatives, makes the equation true for every value of x in that interval.
That definition is also a procedure, and it is the most useful habit in Module 2: to check an answer, substitute it back. Every worked solution in this book has been checked that way by machine, and you can check yours by hand in a minute.
Worked, so the procedure is concrete. Is y = e to the 2x a solution of dy/dx = 2y?
Its derivative is 2 e to the 2x. And 2y is 2 e to the 2x. The two agree, so yes.
dy/dx = 2y
y = C e^(2x)
That block is the machine performing exactly that check, for every value of C at once.
Is y = x squared a solution of the same equation? Its derivative is 2x, and 2y is 2x squared. Those are not equal, so no.
2x = 2x^2
The simplest differential equation of all
dy/dx = f(x)
That says: y is a function whose derivative is f. So y is the integral of f, plus a constant.
Every differential equation is, at bottom, an integration problem in disguise, and the whole of Module 2 is a set of techniques for getting an equation into a form where an integration can actually be done.
What a Differential Equation Is
That is also where the arbitrary constant comes from: an integration always brings one.
Where they come from
A differential equation is what you write down when you know how something changes but not what it is.
| Situation | What you know | The equation |
|---|---|---|
| Radioactive decay | the rate of decay is proportional to the amount left | dN/dt = -kN |
| Cooling | the rate of cooling is proportional to the excess temperature | dT/dt = -k(T - room) |
| A charging capacitor | the current is proportional to the missing voltage | dV/dt = (E - V)/RC |
| Population with limits | growth is proportional to size and to room left | dP/dt = kP(M - P) |
| A falling body with drag | acceleration is gravity minus drag | dv/dt = g - kv |
In each row the left column is the thing you want and the middle column is what physics or observation gives you. The equation is the bridge, and solving it is crossing.
The chapter on where differential equations come from does four of these in full.
Why this module comes after the transform half
It does not have to, and MU prints them in this order for her own reasons. But there is a connection worth seeing from the start.
Module 1's second half solved differential equations by carrying them into the s domain, doing algebra, and carrying the answer back. That method works only for linear equations with constant coefficients, and only when initial values are given at t = 0.
Module 2 solves equations that are not linear, that have coefficients depending on x, and that have no initial values attached. It is the more general set of tools, and it is the one you need when the transform method does not apply. The two halves are complementary, not repetitive.
Check yourself
Which of these are differential equations, and what is the unknown?
| Equation | Differential | Unknown |
|---|---|---|
| dy/dx + 3y = 0 | yes | the function y |
| x squared plus 3x = 4 | no | the number x |
| y prime prime plus y = sin x | yes | the function y |
| dy/dx = 5 | yes | the function y |
And verify these solutions by substitution.
dy/dx = 5
y = 5x + C
dy/dx + y = 0
y = C e^(-x)
d2y/dx2 + y = 0
y = C1 cos(x) + C2 sin(x)
The last one has two arbitrary constants, because the equation is of second order. The next chapter says why that is always so.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself for the same subject.