Lesson 4.10a-c

4.10a-c First order differential equations and modelling Quiz: OCR Further Maths, Unit 1

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson 4.10a-c, First order differential equations and modelling: 20 multiple choice questions for the OCR Further Maths (H245), Unit 1: Pure Core (Y540, Y541), written with Revision Ninja.

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The 20 questions

  1. What is the integrating factor for a differential equation of the form dy/dx + P(x)y = Q(x)?

    • e^(P(x))
    • e^(∫Q(x)dx)
    • e^(∫P(x)dx)
    • ∫P(x)dx

  2. What type of differential equation can be solved directly using an integrating factor?

    • Second-order non-linear
    • Second-order linear
    • First-order linear
    • First-order non-linear

  3. What is the integrating factor for the differential equation dy/dx + 3y = e^x?

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

  4. What is the integrating factor for the differential equation dy/dx + (1/x)y = x^2?

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

  5. What substitution is used to simplify a homogeneous first-order differential equation dy/dx = f(y/x)?

    • y = v + x
    • y = vx
    • y = v/x
    • x = vy

  6. Using the substitution y = vx, what expression replaces dy/dx in a differential equation?

    • x(dv/dx)
    • v(dx/dv)
    • v + x(dv/dx)
    • v + dv/dx

  7. What is the general solution of the differential equation dy/dx = 4y?

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

  8. What is the integrating factor for the differential equation dy/dx - 2y = 5?

    • 5e^(-2x)
    • e^(-2x)
    • e^(2x)
    • -2x

  9. What is the general solution to the first-order differential equation dy/dx + y = 0?

    • y = A e^x
    • y = A e^(-x)
    • y = -x + c
    • y = A - x

  10. Which differential equation form can be solved by separating variables?

    • dy/dx = f(x)g(y)
    • dy/dx = f(x) + g(y)
    • dy/dx = f(x/y)
    • d^2y/dx^2 = f(x)

  11. What differential equation models Newton's Law of Cooling for temperature T and ambient temperature T0?

    • dT/dt = k(T + T0)
    • d^2T/dt^2 = -k(T - T0)
    • dT/dt = -k(T - T0)
    • dT/dt = -kT0

  12. What is the integrating factor for the differential equation dy/dx + tan(x)y = 1?

    • sec(x)
    • ln(cos x)
    • cos(x)
    • sin(x)

  13. What is the integrating factor for the differential equation dy/dx + (2/x)y = 4x?

    • x^2
    • 2 ln(x)
    • 2x
    • x^(-2)

  14. In differential equation modelling, what does an initial condition like y(0) = 3 determine?

    • Particular solution
    • Integrating factor
    • Order of equation
    • General solution

  15. If the integrating factor I(x) = e^(2x), what does d/dx [I(x)y] equal?

    • 2e^(2x)y
    • e^(2x)(dy/dx + y)
    • e^(2x)(dy/dx + 2y)
    • e^(2x)(dy/dx)

  16. A quantity x increases at a rate proportional to x. Which differential equation represents this?

    • dx/dt = k/x
    • dx/dt = k + x
    • d^2x/dt^2 = kx
    • dx/dt = kx

  17. What is the general solution of the differential equation dy/dx = 3x^2?

    • y = x^3 + c
    • y = 6x + c
    • y = x^3
    • y = 3x^3 + c

  18. What is the integrating factor for the differential equation dy/dx - (1/x)y = x?

    • 1/x
    • x
    • -ln(x)
    • -x

  19. Water enters a tank at 5 litres per minute and leaves at 2% of current volume V. What is dV/dt?

    • dV/dt = 5 - 0.02V
    • dV/dt = 5 + 0.02V
    • dV/dt = 5V - 0.02
    • dV/dt = 0.02V - 5

  20. What is the order of the differential equation dy/dx + x y^3 = sin(x)?

    • First order
    • Third order
    • Zero order
    • Second order

All OCR Further Maths quizzes