Second-Order Linear Equations

Two derivatives, two constants, two basic solutions.

12 problems · about 23 minutes · F-IF.B.6, A-CED.A.1

What this lesson teaches

The student states the structure of second-order linear equations and their general solution.

A second-order linear homogeneous equation has a general solution c₁y₁ + c₂y₂.
Superposition means any combination of solutions is again a solution.
Two initial conditions are needed, typically the starting position and velocity.

Straightforward practice. Get the method working first.

5 problems

A fourth-order linear equation. How many arbitrary constants does the general solution have?
Answer 4
Why 4.
What does superposition say?
Answer Any combination of solutions is itself a solution.
Why Combinations stay solutions.
A third-order equation. How many arbitrary constants?
Answer 3
Why One per order.
y = A cos t + B sin t with y = 5 and y′ = 0 at t = 0. What is A?
Answer 5
Why cos 0 is 1.
The same solution with y = 5 and y′ = 0 at t = 0. What is B?
Answer 0
Why The derivative at zero picks out B.

The same ideas with more to keep track of.

3 problems

Two solutions where one is triple the other. Are they independent? 1 yes, 0 no.
Answer 0
Why A multiple adds nothing.
y₁ and y₂ both solve a linear homogeneous equation. Does y₁ + y₂? 1 yes, 0 no.
Answer 1
Why Superposition.
How many independent solutions does a second-order equation need?
Answer 2
Why One per order.

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

4 problems

Sort each statement by whether it holds for a linear homogeneous equation.
Answer True: The sum of two solutions is a solution, A multiple of a solution is a solution · False: The product of two solutions is a solution, One initial condition determines the answer
Why Linearity covers sums and multiples, nothing more.
Is y = 0 always a solution of a homogeneous linear equation? 1 yes, 0 no.
Answer 1
Why Everything on the left becomes zero.
The Count: A second-order equation. How many arbitrary constants does its general solution have?
Answer 2
Why 2.
The Conditions: How many initial conditions are needed to pin down a second-order solution?
Answer 2
Why 2.