What it is
A differential equation is an equation whose unknown is a function and which involves that function’s derivative. A solution is a function that makes the equation true for every x when it is substituted. The order is the highest derivative in the equation: y′ = 2y has order 1 and y″ + y = 0 has order 2.
Three examples with their solutions:
- y′ = 3x². The solutions are y = x³ + C. The derivative of x³ + C is 3x² whatever the value of C.
- y′ = 2y. The solutions are y = Ce^(2x). For C = 3, y = 3e^(2x) and y′ = 6e^(2x) = 2y.
- y″ = −y. The solutions are y = A cos x + B sin x. For y = cos x, y′ = −sin x and y″ = −cos x = −y.
Each example has a whole family of solutions, because the constants C, A and B can be chosen freely. An initial condition gives the value of y, and for second order also of y′, at one point. A first-order equation needs one condition and a second-order equation two, and with them one solution is picked out: y′ = 2y and y(0) = 3 give 3 = Ce⁰ = C, so y = 3e^(2x).
Not differential equations:
- x² − 5x + 6 = 0. The unknown is a number, x = 2 or x = 3, and no derivative is involved.
- y = 2x + 1. This is a function, not an equation involving its derivative.
Differential equations describe processes where the rate of change depends on the quantity itself or on time:
| Process | Differential equation | Solution |
|---|---|---|
| Something grows by 3% of the amount per unit of time | y′ = 0.03y | y = Ce^(0.03x) |
| Coffee at 90 °C cools in a 20 °C room, with Newton’s law of cooling as the model (t in minutes) | T′ = −0.1(T − 20) | T = 20 + 70e^(−0.1t) |
| A radioactive substance decays | N′ = −kN, k > 0 | N = N₀e^(−kt) |
| Free fall without air resistance | v′ = g | v = v₀ + gt |
According to the model, the coffee is at 20 + 70e^(−1) ≈ 45.8 °C after 10 minutes.
Verifying a solution. Often the task is not “solve” but “show that y = … is a solution”. Differentiate y, substitute y and its derivatives into the equation, and see whether the two sides agree for every x.
- Is y = 5e^(3x) a solution of y′ = 3y? y′ = 15e^(3x) and 3y = 15e^(3x). The sides agree, so yes.
- Is y = x² a solution of y′ = 2y? y′ = 2x but 2y = 2x². The sides agree at x = 0 and x = 1 but not for every x, so no.
- Is y = sin 2x a solution of y″ + 4y = 0? y′ = 2 cos 2x and y″ = −4 sin 2x, so y″ + 4y = −4 sin 2x + 4 sin 2x = 0. Yes.
Why it matters
The processes in the table have one thing in common: you know how something changes and want to know what it looks like over time. The types with pages of their own:
- y′ = f(x) is solved with an antiderivative.
- y′ = ky gives exponential growth and decay, y = Ce^(kx).
- Separable differential equations are solved by gathering y on one side and x on the other and integrating.
- First-order linear differential equations, y′ + ay = f(x), are solved with an integrating factor.
- Second-order and nonhomogeneous linear equations describe oscillations, among other things.
- Bounded growth and logistic growth are models whose differential equations have solutions involving e.
Where no formula for the solution exists, it is computed in small steps, as in Euler’s method.
Common misconceptions
- “A differential equation is solved like an ordinary equation, by moving terms and dividing.” The answer is a function, or a family of functions, not a number. y′ = 2y is not solved by dividing by y′.
- “You integrate both sides as they stand.” y′ = 2y does not give y = 2xy + C. The integral of 2y with respect to x is not 2xy, because y is a function of x, not a constant. How to separate y and x is shown on the separable differential equations page.
- “The constant goes on at the end.” For y′ = 2y, y = Ce^(2x) is right and y = e^(2x) + C is wrong. With C = 1: y = e^(2x) + 1 has y′ = 2e^(2x), but 2y = 2e^(2x) + 2.
- “There is only one solution.” Without a condition there is a whole family: y′ = 2y has a solution for every value of C. The condition y(0) = 3 picks out y = 3e^(2x).
- “The solution can always be written as a formula.” Many differential equations have solutions that cannot be written with the common functions, and those are computed numerically.
Common questions
What is the difference between a general and a particular solution?
The general solution is the whole family with its constants, such as y = Ce^(2x). A particular solution is one
function from the family, such as y = 3e^(2x), picked out by an initial condition. For a linear differential
equation the general solution has as many constants as the order.
How do I see the order?
Look at the highest derivative: y′ means first order and y″ second. y′ + y² = x is first order despite the
square, because the order counts derivatives, not powers.
How do I verify that a function is a solution?
Differentiate the function, substitute it and its derivatives into the equation and simplify. If the left side
equals the right side for every x, the function is a solution. An initial condition is checked by substituting
the given x-value.