Second-Order Linear Equations

Two derivatives, two constants, two basic solutions.

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

NameDate

Figure — use these to answer the problems

y″ + y = 0 has sine and cosine as its two basic solutions. Every solution is a combination of these.

y = 2 sin(1x) + 0

the amplitude2

the frequency1

the time shown1

Figure 1

Straightforward practice. Get the method working first.

5 problems

A fourth-order linear equation. How many arbitrary constants does the general solution have?
Answer
2
What does superposition say?
- Any combination of solutions is itself a solution.
- The product of two solutions is a solution.
A third-order equation. How many arbitrary constants?
Answer
y = A cos t + B sin t with y = 5 and y′ = 0 at t = 0. What is A?
Answer
The same solution with y = 5 and y′ = 0 at t = 0. What is B?
Answer

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
y₁ and y₂ both solve a linear homogeneous equation. Does y₁ + y₂? 1 yes, 0 no.
Answer
How many independent solutions does a second-order equation need?
Answer

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.
Write each item under the heading it belongs to: The sum of two solutions is a solution · A multiple of a solution is a solution · The product of two solutions is a solution · One initial condition determines the answer
True
False
Is y = 0 always a solution of a homogeneous linear equation? 1 yes, 0 no.
Answer
The Count
A second-order equation. How many arbitrary constants does its general solution have?
Answer
The Conditions
How many initial conditions are needed to pin down a second-order solution?
Answer