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

The 20 questions

1. Which pair of equations is a coupled system of first order linear differential equations?

   • dx/dt = x^2 + y and dy/dt = x, which is a nonlinear coupled system
   • d^2x/dt^2 + x = 0, which involves only one dependent variable and no coupling
   • dx/dt = ax + by + f(t) and dy/dt = cx + dy + g(t)
   • dy/dx = xy and dx/dt = 1, which is nonlinear and separable only

2. What is the main reason to model predator-prey populations with coupled differential equations?

   • The model has only one dependent variable, the population of prey alone
   • Both populations are constant at all times because the system is in balance
   • Each population grows independently at a fixed rate that never changes
   • Each population's rate of change depends on the other population

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. For x' = 2x - y and y' = x, what are the equilibrium points?

   • (1, 2)
   • (0, 0) only
   • All points on the line y = 2x
   • No equilibrium points

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 x' = -2x and y' = -y, what happens as t tends to infinity?

    • Both x and y grow without bound
    • Both x and y tend to 0
    • x and y are constant
    • x grows and y decays

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. In a steady state of a predator-prey model, what must hold?

    • x + y = 0 only
    • The eigenvalues equal zero
    • x = y = t
    • dx/dt = 0 and dy/dt = 0

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. In a coupled linear system, which step eliminates one variable to get a single second-order equation?

    • Integrate both equations with respect to t at once
    • Add the two equations and set the sum to zero
    • Differentiate one equation and substitute the other
    • Divide one equation by the other only

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. The eigenvalues of a coupled linear system are +/- 2i. What is the behaviour of the solutions?

    • Periodic oscillation with period pi
    • Steady state at zero
    • Exponential growth with rate 2
    • Exponential decay with rate 2

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 the predator-prey model dy/dt = -cy + dx with c > 0, what happens to y if there is no prey, so x = 0?

    • y decays exponentially to zero
    • y grows exponentially without any limit over time
    • y stays constant at its initial value forever
    • y oscillates forever with a fixed constant amplitude

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