Systems of Differential Equations

Two quantities changing, each depending on the other.

12 problems · about 23 minutes · A-CED.A.3, N-VM.C.11

What this lesson teaches

The student writes systems of first-order equations and converts second-order equations into them.

A system has several unknowns whose rates depend on each other, so they cannot be solved separately.
A linear system is written x′ = Ax, which opens all of linear algebra.
Any higher-order equation becomes a first-order system by naming the derivatives as new unknowns.

Straightforward practice. Get the method working first.

5 problems

A fifth-order equation converted to a system. How many first-order equations?
Answer 5
Why 5.
Why convert a higher-order equation into a system?
Answer Numerical software solves first-order systems, so everything is converted first.
Why First-order systems are what solvers accept.
A third-order equation converted to a system. How many first-order equations?
Answer 3
Why One per order.
y″ + 5y = 0 with v = y′. What is the coefficient of y in the equation for v′?
Answer -5
Why Move the 5y across.
x′ = 2x + 3y and y′ = x. Is the system coupled? 1 yes, 0 no.
Answer 1
Why Each rate involves both unknowns.

The same ideas with more to keep track of.

3 problems

x′ = 2x and y′ = 5y. Is the system coupled? 1 yes, 0 no.
Answer 0
Why They can be solved separately.
A system of two equations written as x′ = Ax. How many rows does A have?
Answer 2
Why One per equation.
x′ = 3x + y with x = 2 and y = 4. What is x′?
Answer 10
Why 6 + 4.

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

4 problems

Order the steps of converting a second-order equation to a system.
Answer 1. Introduce a new unknown v equal to the first derivative 2. Write the first equation as y prime equals v 3. Rearrange the original to isolate the second derivative 4. Write that as the equation for v prime
Why The new unknown must be named first.
y′ = 4x − y with x = 3 and y = 2. What is y′?
Answer 10
Why 12 − 2.
The Conversion: A fourth-order equation converted to a first-order system. How many equations does the system have?
Answer 4
Why 4.
The State: A second-order equation. How many numbers make up the state vector?
Answer 2
Why 2.