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Cogito

Differential Equations · Chapter 7 · Lesson 1

Practice Worksheet

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

NameDate

Figure — use these to answer the problems

Each arrow shows how both quantities change at once. Release a state and watch the pair evolve together.
  • Released from(2, 1)
the coupling strength1
starting first quantity2
starting second quantity1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. A fifth-order equation converted to a system. How many first-order equations?

    Answer

  2. Why convert a higher-order equation into a system?

    • Numerical software solves first-order systems, so everything is converted first.
    • The system is always shorter to write.

  3. A third-order equation converted to a system. How many first-order equations?

    Answer

  4. y″ + 5y = 0 with v = y′. What is the coefficient of y in the equation for v′?

    Answer

  5. x′ = 2x + 3y and y′ = x. Is the system coupled? 1 yes, 0 no.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. x′ = 2x and y′ = 5y. Is the system coupled? 1 yes, 0 no.

    Answer

  2. A system of two equations written as x′ = Ax. How many rows does A have?

    Answer

  3. x′ = 3x + y with x = 2 and y = 4. What is x′?

    Answer

Stretch Yourself

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

4 problems

  1. Order the steps of converting a second-order equation to a system.

    Write 1 to 4 in the boxes to put these in order.

    • Introduce a new unknown v equal to the first derivative
    • Write the first equation as y prime equals v
    • Rearrange the original to isolate the second derivative
    • Write that as the equation for v prime

  2. y′ = 4x − y with x = 3 and y = 2. What is y′?

    Answer

  3. The Conversion

    A fourth-order equation converted to a first-order system. How many equations does the system have?

    Answer

  4. The State

    A second-order equation. How many numbers make up the state vector?

    Answer