Lesson · Differential Equations and Euler's Method · What a differential equation is

What a Differential Equation Is

An equation in algebra asks for a number. A differential equation asks for a function: it says how fast the unknown changes and leaves you to find what the unknown is. Its order is the highest derivative it mentions, and a proposed solution is tested by substituting it and reading the residual, the left side minus the right. The first surprise is that a solvable equation usually has infinitely many answers, not one.

Open the interactive lesson Practice

y′ = 2t the unknown is the function y order = the highest derivative present residual = left side − right side residual 0 for every t: a solution y = t² and y = t² + 3 both work

A differential equation, its order, and the residual test

Core ideas

What this lesson establishes

Three ideas, in the order you need them: what is being asked for, how to size the equation, and how to decide whether an answer is one.

The unknown is a function, not a number

In 3x + 1 = 7 the unknown is a number and there is one right value. In y′ = 2t the unknown is a function y of t, and the equation does not say what y is. It says what y does: at every t, the rate of change of y is 2t. To answer it you must produce a whole function, and to check an answer you must check it at every t at once.

The order is the highest derivative that appears

y′ = 2t is first order, because only the first derivative appears. y″ + y = 0 is second order, because y″ does. The order tells you how much extra information a single answer will need, which “Initial Value Problems” makes exact. It is read from the equation and nothing else: the powers of y and the other terms do not change it.

A solution is checked by substituting it and reading the residual

Differentiate the candidate, put it and its derivatives into the equation, and subtract the right side from the left. That difference is the residual. If it is 0 for every t, the candidate is a solution. If it is anything else, it is not, and the residual says by how much and where it fails. The check is algebra on a formula, not a trial at a few values.

The material

An equation about a rate, and what counts as an answer

The definition, the order, the residual test on three candidates, and the first look at why there are many solutions.

Definition · Differential equation, order, solution

A differential equation is an equation whose unknown is a function and which involves one or more of that function's derivatives. Its order is the highest derivative that appears. A solution is a function that makes the equation true for every t in some interval.

The word “every” carries the weight. A function that fits the equation at a few values of t has satisfied a few instances of it, and no more.

Start with the smallest example. The equation y′ = 2t says that the rate of change of y at time t is 2t: zero at the start, 2 at time 1, 4 at time 2. In Accumulation and the Integral you met this question the other way up, as “which function has this rate”, and the answer was an antiderivative. A differential equation is that question asked of a rate that may also depend on the unknown itself, as in y′ = y, and then the antiderivative alone no longer answers it.

The lab below takes an equation and a candidate and does the substitution for you, symbolically. It differentiates the candidate exactly, forms y′ − 2t, simplifies, and prints the result in the tile named Residual. Nothing is evaluated at sample points; the verdict is about every t.

equation y′ = 2t candidate its y′ residual y′ − 2t verdict t² 2t 0 a solution t² + 3 2t 0 a solution t³ 3t² 3t² − 2t not a solution

Read the three rows. The candidate t² has derivative 2t, so the residual 2t − 2t is 0. The candidate t² + 3 has the same derivative, because a constant has rate zero, so it is a solution as well. The candidate t³ has derivative 3t², and the residual 3t² − 2t is not zero. It does vanish at t = 0 and at t = 2/3, and that does not rescue it: a solution needs the residual to be zero at every t.

One equation, many answers

That t² and t² + 3 are both solutions is the first fact to absorb, and the one the misconception below gets wrong. Adding any constant C to t² leaves the derivative at 2t, so t² + C is a solution for every C. A differential equation of this kind has a whole family of solutions, one curve for each constant, and each is the same shape shifted up or down.

Example · The same check on a candidate that does not work

Take y′ = 2t and the candidate y = 2t. Its derivative is 2, so the residual is 2 − 2t, which the lab writes as −2t + 2.

That is zero at t = 1 and nowhere else. At t = 1 the candidate and the equation agree, and one instance of agreement is exactly what the residual test refuses to accept. “Checking a Proposed Solution” meets the same trap in a harder equation.

What the lab does and does not claim

The residual tile is exact: it is the symbolic left side minus right side, and it prints 0 only when the simplified expression is identically zero. The curve in the plot is a drawing made with ordinary floating-point numbers, and it is there to be looked at, not to be trusted for a digit.

Interactive

Checking a solution

Substitute, simplify, compare with zero

Substitute a candidate and read the residual

Choose each preset in turn. The order tile reads the equation, the residual tile is the left side minus the right after substitution, and the verdict follows from whether that is zero for every t. Then type your own candidate for y′ = 2t, such as t^2 + 5 or 2t, and predict the residual before you look.

Equation and candidate

Equation, with y' and y'' for the derivatives

Candidate y(t)

Initial values: t0 y0, or t0 y0 v0 (may be empty)

Order—

Linear?—

Residual—

Verdict—

Initial values—

Constants—

Candidates: a ratio of polynomials in t, such as 1/(1 - t), or sums of terms such as 3t^2, 2e^(-t), t cos(2t) and C e^(3t), with the constants C and D.

Method

Testing a candidate against an equation

The same four moves for every equation on this course, in the same order.

Name the unknown and read the order

Find the unknown function and the highest derivative of it in the equation. That is the order, and it is all the order is.

Move everything to one side

Write the equation as left side minus right side. For y′ = 2t that is y′ − 2t. The candidate will be tested against zero, so the zero has to be on the right.

Differentiate the candidate and substitute

Compute every derivative the equation mentions, exactly, and put the candidate and those derivatives where y and its derivatives stand.

Simplify and read the residual

If it simplifies to 0 for all t, the candidate is a solution. If any t remains in it, it is not, and the residual tells you what is left over.

Order, residual and solution

Score 0 / 0

Choose the answer you can justify.

Two candidates for y′ = 2t

Test y = t³ and y = t² + 3 against y′ = 2t, with the residual taken as the derivative minus 2t.

y = t³: y′ = 3t² residual = 3t² − 2t = t·(3t − 2) not zero for all t, so not a solution y = t² + 3: y′ = 2t + 0 = 2t residual = 2t − 2t = 0 a solution y = t² + C, for any constant C: y′ = 2t, residual 0, so a solution for every C

The third candidate is the one to keep. The constant 3 dropped out of the derivative, and so would any other constant, which is why the equation has one solution for every C and not a single answer. A single curve will appear only when something fixes C, and “Initial Value Problems” is where that happens.

The factored residual t·(3t − 2) shows why t³ is no solution even though the equation holds at two points: it is zero at t = 0 and t = 2/3 and nonzero between and beyond them.

Common mistakes

Expecting a differential equation to have a number for an answerThe unknown is a function, and y′ = 2t is satisfied by t² and by t² + 3 and by t² − 7, each with residual zero in the lab. There is no single value to find and no single curve. A number comes out only if you are asked for one value of a particular solution, after something has chosen which curve.

Reading the order from the exponent or the number of termsThe order is the highest derivative and nothing else. y′ = y² is first order even though it has a square in it, and y″ = 0 is second order even though it is short.

Testing a candidate at one value of t and stoppingThe residual of t³ in y′ = 2t is zero at t = 0, so a check there passes and the candidate is wrong. The test is the simplified residual for every t, which is exactly what the lab prints.

Check your understanding

✓

Finish when you can say what a differential equation is, name its order, and decide a candidate.

You should be able to state in a sentence that the unknown is a function, read the order of an equation from its highest derivative, substitute a candidate and simplify the residual by hand on a first-order equation, and decide whether it is a solution from whether that residual is zero for every t. You should also be able to say why t² and t² + 3 are both answers to the same equation.

Note: This lesson tested candidates that were handed over. The next uses the same residual on harder equations, a second-order one with an exponential, one with a coefficient that depends on t, and one that is not linear, and learns to tell which kind it is. “Checking a Proposed Solution” is where.