Lesson I1-I3

I1-I3 First order equations with integrating factors and modelling Quiz: AQA Further Maths, Unit 2

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson I1-I3, First order equations with integrating factors and modelling: 20 multiple choice questions for the AQA Further Maths (7367), Unit 2: Compulsory content, written with Revision Ninja.

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The 20 questions

  1. For the first order equation dy/dx + P(x) y = Q(x), what is the integrating factor?

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

  2. Multiplying dy/dx + P(x) y = Q(x) by an integrating factor I(x) gives which equation?

    • I dy/dx = Q
    • d/dx (I y) = Q
    • d/dx (y) = I Q
    • d/dx (I y) = I Q

  3. A general solution of a first order differential equation contains how many arbitrary constants?

    • Two
    • One
    • None
    • Depends on the value of x

  4. How is a particular solution obtained from a general solution?

    • By differentiating the general solution twice
    • By adding a second general solution
    • By substituting a given initial or boundary condition to find the constant
    • By setting the constant to zero always

  5. What is the integrating factor for dy/dx - 3y = x?

    • e^(-3x)
    • -3 e^(-3x)
    • e^x
    • e^(3x)

  6. Which equation is a first order linear differential equation?

    • dy/dx + y^2 = x
    • d^2y/dx^2 + y = 0
    • dy/dx + 3xy = sin(x)
    • y dy/dx = x

  7. When is the integrating factor method appropriate for solving a differential equation?

    • Only when the equation is separable
    • When the right-hand side is always zero
    • When the equation is linear in y and first order
    • When the equation contains y squared

  8. Solve dy/dx + y = e^x with y(0) = 0.

    • y = e^x / 2
    • y = e^x - e^(-x)
    • y = sinh(x)
    • y = cosh(x)

  9. Find the general solution of dy/dx - 2y = 1.

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

  10. Find the general solution of dy/dx + 2xy = 2x.

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

  11. A particle has dv/dt + v/5 = 10 with v(0) = 0. Which expression gives v in terms of t?

    • v = 50(1 - e^(-t/5))
    • v = 10(1 - e^(-t/5))
    • v = 50 e^(-t/5)
    • v = 50(1 - e^(t/5))

  12. In the model dv/dt + v/5 = 10 with v(0) = 0, what is the limiting value of v as t tends to infinity?

    • 10
    • 0
    • 5
    • 50

  13. Find the general solution of dy/dx + 2y/x = 4x for x > 0.

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

  14. Solve dy/dx + y = 0 with y(0) = 3.

    • y = -3 e^(-x)
    • y = 3 - x
    • y = 3 e^(-x)
    • y = 3 e^x

  15. Find the general solution of dy/dx + y = x.

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

  16. Find the general solution of dy/dx + y/x = x^2 for x > 0.

    • y = x^3/4 + C/x
    • y = x^3/4 + C
    • y = x^3/3 + C/x
    • y = x^2/4 + C/x

  17. Find the particular solution of dy/dx + 2y = 4 with y(0) = 3.

    • y = 3 e^(-2x)
    • y = 4 + e^(-2x)
    • y = 2 - e^(-2x)
    • y = 2 + e^(-2x)

  18. What is the integrating factor for dy/dx + (2/x) y = sin(x) when x > 0?

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

  19. For dy/dx + y tan(x) = sec(x), find the general solution for -pi/2 < x < pi/2.

    • y = sin(x) + C sec(x)
    • y = cos(x) + C sin(x)
    • y = tan(x) + C cos(x)
    • y = sin(x) + C cos(x)

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

    • y = 4x e^(2x)
    • y = 4 e^(2x)
    • y = 2 e^(4x)
    • y = 4 e^(-2x)

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