What a Differential Equation Is

An equation whose unknown is a whole function.

12 problems · about 22 minutes · F-IF.B.6, A-CED.A.1

What this lesson teaches

The student identifies differential equations, their order, and verifies solutions.

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.

Straightforward practice. Get the method working first.

5 problems

y″ + 4y′ + 3y = 0. What is the order?
Answer 2
Why 2.
What is the unknown in a differential equation?
Answer A whole function, described by how it changes.
Why A function.
dy/dx = x². What is the order?
Answer 1
Why Only a first derivative appears.
y‴ − y = 0. What is the order?
Answer 3
Why Three primes.
dy/dt = 3y with y = 7. What is dy/dt?
Answer 21
Why 3 × 7.

The same ideas with more to keep track of.

3 problems

dy/dx = 2y and y = 0. What is dy/dx?
Answer 0
Why A constant zero solution.
Is y = 5 a solution of dy/dx = 0? 1 yes, 0 no.
Answer 1
Why A constant has zero derivative.
How many arbitrary constants does the general solution of a first-order equation usually have?
Answer 1
Why One per order.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each equation by its order.
Answer First order: dy/dx = 4y, dy/dt + 2y = t · Second order: y″ + y = 0, y″ − 3y′ = 5
Why Look for the highest derivative in each.
An initial value problem for a first-order equation. How many solutions does it usually have?
Answer 1
Why The starting value picks one curve.
The Order: y″ + 3y′ − y = 0. What is the order of this equation?
Answer 2
Why 2.
The Rate: dy/dt = 5y and y = 4 at the moment asked. What is dy/dt then?
Answer 20
Why 20.