An ordinary equation asks for a number. A differential equation asks for a function, described by how it changes. Its order is the highest derivative present. To verify a candidate, substitute it and its derivatives and see whether both sides agree for all x. A first-order equation has a family of solutions, one per constant, and an initial condition selects one member.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why they are everywhere
You rarely know a quantity directly, but you often know the rule for its rate of change: a population grows in proportion to its size, a cup cools in proportion to how hot it is. That rule is a differential equation.
Order
dy/dx = 2y is first order. y″ + y = 0 is second order: y = sin x works, since y″ = −sin x and −sin x + sin x = 0. The order says how many constants the general solution will carry.
A candidate that fails
Is y = x² a solution of dy/dx = 2y? dy/dx = 2x, but 2y = 2x². Equal only at x = 0 and x = 1, not for every x. Not a solution. Agreement at one point is not enough.
A family
y = Ce^(−3x) + 2 solves y′ + 3y = 6 for every C: y′ = −3Ce^(−3x), and −3Ce^(−3x) + 3Ce^(−3x) + 6 = 6. One equation, infinitely many solutions, one per value of C.
An initial condition selects one
Add y(0) = 5: C + 2 = 5, so C = 3 and y = 3e^(−3x) + 2. An equation with a starting value is an initial value problem, and it typically has exactly one solution.
Read the solution
y = 3e^(−3x) + 2 starts at 5 and settles toward 2 as x grows. The equation y′ = 6 − 3y says the same: the rate is zero exactly at y = 2. Reading the equation predicts the solution before solving.
Step 2: Try It Yourself
Tap and try it out.
Verify that y = Ce^(−3x) + 2 solves y′ + 3y = 6 for every C, state the order, then apply y(0) = 5 and check
- 1y′ = −3Ce^(−3x)differentiate the candidate
Step 0 of 4
- The equationdy/dx = a·y
- Through(0, 1)
Every short line shows the slope the equation demands at that point. The curve simply follows them, and the marked point is the initial condition that picks it out of the family.
Step 3: In Real Life
A skydiver
A skydiver’s speed changes at a rate that depends on the speed itself: air resistance grows with it. The unknown is the whole speed history, not one number. That is a differential equation.
Step 4: Watch an Example
One step at a time.
Watch Ravi Verify a Solution
Ravi checks whether y = 3e²ˣ solves dy/dx = 2y.
- Step 1
He differentiates the candidate, getting dy/dx = 6e²ˣ, and computes the right-hand side, 2y = 2 × 3e²ˣ = 6e²ˣ.
Step 5: Your Turn
Practice makes it stick.
The Order
Problem 1 of 2
y″ + 3y′ − y = 0. What is the order of this equation?
The Rate
Problem 2 of 2
dy/dt = 5y and y = 4 at the moment asked. What is dy/dt then?
Read the Equation
1 of 8
dy/dx = x². What is the order?
2 of 8
y‴ − y = 0. What is the order?
3 of 8
dy/dt = 3y with y = 7. What is dy/dt?
4 of 8
dy/dx = 2y and y = 0. What is dy/dx?
5 of 8
Is y = 5 a solution of dy/dx = 0?
6 of 8
How many arbitrary constants does the general solution of a first-order equation usually have?
7 of 8
Sort each equation by its order.
Tap something to move it.
- Empty
- Empty
8 of 8
An initial value problem for a first-order equation. How many solutions does it usually have?
Step 6: Quick Check
Show what you know.
Question 1 of 2
y″ + 4y′ + 3y = 0. What is the order?
Question 2 of 2
What is the unknown in a differential equation?
What You Learned
- A differential equation describes an unknown function by how it changes.
- Its order is the highest derivative that appears, and says how many constants to expect.
- Verify a candidate by substituting it and comparing both sides for every x.
- A first-order equation has a family of solutions, and an initial condition selects one.