9.9 Coupled first order differential equations Quiz: Pearson Edexcel Further Maths, Unit 9

The 20 questions

1. How many dependent variables are in a pair of coupled first-order linear differential equations?

   • Three
   • One
   • Two
   • Four

2. Why are predator-prey models expressed as coupled differential equations?

   • Independent growth rates
   • Zero death rates
   • Constant total population
   • Interdependent growth rates

3. For x' = x + y and y' = 4x + y, what are the eigenvalues of the coefficient matrix?

   • 3 and -1
   • -3 and 1
   • 2 and 2
   • 1 and 4

4. What is the general solution for x(t) of x' = x + y and y' = 4x + y?

   • x = A e^(-3t) + B e^t
   • x = A e^t + B e^(4t)
   • x = A e^(3t) + B e^(-t)
   • x = (A + Bt) e^t

5. For x' = -y and y' = x with x(0) = 1 and y(0) = 0, what is x(t)?

   • e^t
   • cos t
   • sin t
   • cosh t

6. What are the eigenvalues of the matrix for x' = -y and y' = x?

   • +/- i
   • +/- 2i
   • +/- 1
   • 0 and 1

7. For x' = x + y and y' = 4x + y, what is y(t) for the eigenvalue -1 part, with x = B e^(-t)?

   • y = B e^(-t)
   • y = 2B e^(-t)
   • y = -2B e^(-t)
   • y = -B e^(-3t)

8. What is the equilibrium point for the system dx/dt = 2x - y and dy/dt = x?

   • (2, 1)
   • (0, 0)
   • (1, 1)
   • (0, 2)

9. A system has x' = 3y and y' = 3x with x(0) = 2 and y(0) = 0. What is x(t)?

   • 2 cosh 3t
   • 2 e^(3t)
   • 2 cos 3t
   • 2 sinh 3t

10. For dx/dt = -2x and dy/dt = -y, what value do x and y approach as t approaches infinity?

    • One
    • Zero
    • Minus infinity
    • Infinity

11. When the system has a constant forcing term, what form is a particular integral normally taken to be?

    • An exponential e^(kt)
    • Zero
    • A constant
    • A linear function of t

12. What is the value of all time derivatives at a steady state in a coupled system?

    • Constant non-zero
    • Infinite
    • One
    • Zero

13. What are the eigenvalues of the matrix [[2, 1], [1, 2]]?

    • 2 and 1
    • 1 and -1
    • 3 and 1
    • 4 and 0

14. What is the sum of the eigenvalues of the matrix [[3, 1], [2, 2]]?

    • 4
    • 6
    • 3
    • 5

15. What operation is performed on a first-order coupled equation to form a second-order equation?

    • Matrix inversion
    • Integration
    • Differentiation
    • Factorisation

16. For x' = 2y and y' = 2x with x(0) = 1 and y(0) = 0, what is x(t)?

    • sinh 2t
    • cosh 2t
    • cos 2t
    • e^(2t)

17. Purely imaginary auxiliary roots in a linear system result in what type of solution behaviour?

    • Periodic oscillation
    • Constant displacement
    • Exponential decay
    • Exponential growth

18. What is the general solution for x' = y and y' = -4x?

    • x = A e^(2t) + B e^(-2t)
    • x = A cos t + B sin t
    • x = A cos 4t + B sin 4t
    • x = A cos 2t + B sin 2t

19. Which single second-order equation is equivalent to the system x' = ax + by, y' = cx + dy?

    • x'' + (a + d) x' + (ad + bc) x = 0
    • x'' - (a + d) x' + (ad + bc) x = 0
    • x'' - (a + d) x' + (ad - bc) x = 0
    • x'' - (a - d) x' + (ad + bc) x = 0

20. In dy/dt = -cy + dx with c > 0, how does y behave if x = 0?

    • Exponential decay
    • Exponential growth
    • Linear growth
    • Periodic oscillation

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