9.1-9.4 First order differential equations Quiz: Pearson Edexcel Further Maths, Unit 9

The 20 questions

1. Which is the integrating factor for dy/dx + P(x) y = Q(x)?

   • integral of P(x) dx
   • e^(integral of P(x) dx)
   • e^(integral of Q(x) dx)
   • e^(-integral of P(x) dx)

2. What is the general solution of dy/dx + 2y = 0?

   • y = A e^(2x)
   • y = A x e^(-2x)
   • y = A/(2x)
   • y = A e^(-2x)

3. Solve dy/dx + y = e^x.

   • y = e^x + C e^(-x)
   • y = e^x + C e^x
   • y = (1/2) e^x + C e^x
   • y = (1/2) e^x + C e^(-x)

4. Solve dy/dx = 2xy with y(0) = 1.

   • y = 2e^(x^2)
   • y = e^(2x)
   • y = x^2 + 1
   • y = e^(x^2)

5. Solve dy/dx = y/x with y(1) = 2.

   • y = 2x
   • y = x + 1
   • y = 2/x
   • y = x^2 + 1

6. Solve dy/dx + y/x = x for x > 0.

   • y = x^2/3 + C/x
   • y = x^2 + C/x
   • y = x^3/3 + C/x
   • y = x^2/3 + C x

7. A particle has dv/dt = -kv with v(0) = 20. What is v(t)?

   • v = 20/(kt)
   • v = 20 - kt
   • v = 20 e^(-kt)
   • v = 20 e^(kt)

8. Newton's law of cooling is dT/dt = -k(T - 20) with T(0) = 100. What is T(t)?

   • T = 20 + 80 e^(-kt)
   • T = 100 e^(-kt) + 20
   • T = 100 e^(-kt)
   • T = 20 + 100 e^(kt)

9. A population satisfies dP/dt = 0.05P with P(0) = 1000. Find P(10) to 1 decimal place.

   • 5000.0
   • 1648.7
   • 1500.0
   • 1050.0

10. What is the integrating factor for dy/dx + (3/x) y = x with x > 0?

    • x^2/3
    • x^3
    • x^(-3)
    • 3x

11. Solve dy/dx + y = 2x with y(0) = 0.

    • y = 2x - 2 - 2e^(-x)
    • y = 2x - 2 + 2e^(-x)
    • y = 2 - 2e^(-x)
    • y = 2x + 2e^(-x)

12. Which method solves dy/dx = x/y, and what is the general solution?

    • Separation giving y^2 = 2x + C
    • Integrating factor giving y = x^2 + C
    • Substitution giving y = Cx
    • Separation of variables giving y^2 = x^2 + C

13. What is the solution of dy/dx = 3y with y(0) = 4, evaluated at x = 1?

    • 4e^3, about 80.3
    • 4 + 3e, about 12.2
    • 12e, about 32.6
    • 4e, about 10.9

14. What is a general solution of dy/dx = 5y?

    • y = A + 5x
    • y = A x^5
    • y = A e^(5x)
    • y = 5A e^x

15. What is needed to turn a general solution into a particular solution?

    • The integrating factor
    • An initial or boundary condition
    • The limiting gradient as x tends to infinity
    • The auxiliary equation

16. What is the integrating factor for dy/dx - (2/x) y = x^2?

    • x^(-2)
    • e^(2x)
    • x^2
    • -2/x

17. A body has dv/dt = 10 - 2v with v(0) = 0. What is v(t)?

    • v = 10(1 - e^(-2t))
    • v = 10t - 2
    • v = 5 e^(-2t)
    • v = 5(1 - e^(-2t))

18. Solve dy/dx + 2xy = 2x with y(0) = 3.

    • y = 2 + e^(-x^2)
    • y = 1 + 3e^(-x^2)
    • y = 1 + 2e^(x^2)
    • y = 1 + 2e^(-x^2)

19. Solve dy/dx = 2y/(x + 1) with y(0) = 1.

    • y = 2(x + 1)
    • y = x^2 + 1
    • y = (x + 1)^2
    • y = e^(2x)

20. A body obeys dT/dt = -0.1(T - 20) with T(0) = 100 in a room at 20. How long does it take to cool to 60?

    • About 6.93 time units
    • About 4.00 time units
    • About 10.0 time units
    • About 13.86 time units

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