Often two quantities change together, each rate depending on both. Predators and prey, two connected tanks, or a circuit with two loops.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Coupling
If each equation involves both unknowns, they cannot be solved separately. The system must be handled as a whole.
Matrix form
A linear system can be written as x′ = Ax, where x is a vector of the unknowns. All of linear algebra becomes available.
Second order becomes a system
Set v = y′. Then y″ + by′ + cy = 0 becomes y′ = v and v′ = −cy − bv, a first-order system.
Why bother
Numerical software solves first-order systems. Every higher-order equation gets converted before it is computed.
The state vector
The pair of position and velocity is the state. Knowing the state fixes the entire future.
Two quantities, each affecting the other
A system couples several unknown functions through their derivatives. Predator and prey populations are the standard example: each rate depends on both quantities.
Linear systems become matrix equations
A linear system is x′ = Ax, with A a matrix of coefficients. That form is why linear algebra is a prerequisite, and it is what makes eigenvalues the natural tool.
High-order equations become systems
Any nth-order equation converts to a system of n first-order ones by naming the derivatives as new variables. That is how numerical solvers handle every equation regardless of order.
Coupling is what makes them interesting
If the equations do not involve each other they can be solved separately. The interaction is what produces oscillation, competition and the rich behaviour systems are studied for.
Step 2: Try It Yourself
Tap and try it out.
- Released from(2, 1)
Every arrow is perpendicular to the line from the origin, so trajectories circle rather than approach.
Step 3: Watch an Example
One step at a time.
Watch Ines Convert an Equation
Ines turns y″ + 3y′ + 2y = 0 into a first-order system.
- Step 1
She introduces a second unknown by setting v = y′.
Step 4: Your Turn
Practice makes it stick.
The Conversion
Problem 1 of 2
A fourth-order equation converted to a first-order system. How many equations does the system have?
The State
Problem 2 of 2
A second-order equation. How many numbers make up the state vector?
Two at Once
1 of 8
A third-order equation converted to a system. How many first-order equations?
2 of 8
y″ + 5y = 0 with v = y′. What is the coefficient of y in the equation for v′?
3 of 8
x′ = 2x + 3y and y′ = x. Is the system coupled? 1 yes, 0 no.
4 of 8
x′ = 2x and y′ = 5y. Is the system coupled? 1 yes, 0 no.
5 of 8
A system of two equations written as x′ = Ax. How many rows does A have?
6 of 8
x′ = 3x + y with x = 2 and y = 4. What is x′?
7 of 8
Order the steps of converting a second-order equation to a system.
- 1Write the first equation as y prime equals v
- 2Rearrange the original to isolate the second derivative
- 3Write that as the equation for v prime
- 4Introduce a new unknown v equal to the first derivative
8 of 8
y′ = 4x − y with x = 3 and y = 2. What is y′?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A fifth-order equation converted to a system. How many first-order equations?
Question 2 of 2
Why convert a higher-order equation into a system?
What You Learned
- 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.