An ordinary equation asks for a number. A differential equation asks for a function, and it describes that function by how it changes.
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. That rule is a differential equation.
Order
The order is the highest derivative that appears. dy/dx = 2y is first order; y″ + y = 0 is second order.
Verifying a solution
Substitute the candidate and its derivatives into the equation. If both sides agree for all x, it is a solution.
Families of solutions
A first-order equation usually has a whole family of solutions, one per constant. An initial condition selects a single member.
Initial value problems
An equation together with a starting value is an initial value problem. It typically has exactly one solution.
The unknown is a whole function
An algebraic equation asks which numbers satisfy it. A differential equation asks which functions do. That is a much larger question, and it is why the answer is usually a family rather than a value.
Order and linearity classify it
The order is the highest derivative present. An equation is linear if the unknown function and its derivatives appear only to the first power and are not multiplied together. Both determine which methods apply.
General and particular solutions
The general solution contains arbitrary constants — one per order. An initial condition fixes each constant, selecting one particular solution from the family.
Verifying a solution is easy
Substitute the candidate and its derivatives into the equation and check both sides agree. Verification is far easier than solving, which makes it a cheap and reliable check on every answer.
Step 2: Try It Yourself
Tap and try it out.
- 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: 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.
Step 4: 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? 1 yes, 0 no.
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 5: 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.
- A first-order equation has a family of solutions, and an initial condition selects one.