Lesson 8.7.1

8.7.1 Solving first order differential equations analytically Quiz: Pearson Edexcel Maths, Unit 8

20 questions

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Lesson 8.7.1, Solving first order differential equations analytically: 20 multiple choice questions for the Pearson Edexcel Maths (9MA0), Unit 8: Integration, written with Revision Ninja.

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The 20 questions

  1. What technique is used to solve first-order differential equations when variables can be completely isolated?

    • Separation of variables
    • Substitution method
    • Partial fractions
    • Integration by parts

  2. When separating variables in dy/dx = f(x)g(y), where should the term g(y) be placed?

    • Numerator on left
    • Denominator on left
    • Left untouched
    • Multiplied on right

  3. What must be included when evaluating an indefinite integral after separating variables?

    • Constant of integration
    • Product rule factor
    • Boundary multiplier
    • Integrating factor

  4. If dy/dx = ky, what family of functions represents the general solution?

    • Exponential functions
    • Linear functions
    • Quadratic functions
    • Trigonometric functions

  5. Solve the differential equation dy/dx = 2x, given that y = 5 when x = 0.

    • y = 2x^2 + 5
    • y = 2x + 5
    • y = x^2 + 2
    • y = x^2 + 5

  6. What type of differential equation requires an integrating factor for analytical solution?

    • First-order linear
    • Separable nonlinear
    • Homogeneous quadratic
    • Second-order linear

  7. What is the correct formula for the integrating factor I(x) for dy/dx + P(x)y = Q(x)?

    • e^(-int P(x) dx)
    • e^(int P(x) dx)
    • int e^(P(x)) dx
    • ln(int P(x) dx)

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

    • 2x
    • x^2
    • ln(x)
    • e^(2x)

  9. What is the result of differentiating the product I(x)y with respect to x during linear ODE solution?

    • I(x)(dy/dx + P(x)y)
    • I(x) dy/dx
    • P(x)I(x)y
    • I'(x) y

  10. Find the general solution to dy/dx = y / x.

    • y = cx
    • y = c / x
    • y = x + c
    • y = cx^2

  11. What is the primary purpose of substituting boundary conditions into a general solution?

    • Change the variable
    • Check the integration
    • Find specific constant
    • Eliminate the derivative

  12. Which substitution is most appropriate for solving dy/dx = (x + y)^2?

    • u = y^2
    • u = y / x
    • u = x + y
    • u = xy

  13. What does the statement y(0) = 4 mean in the context of differential equations?

    • Stationary point
    • Initial condition
    • Asymptote value
    • Boundary derivative

  14. Solve the separable differential equation dy/dx = y^2.

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

  15. If dy/dx = -3y, what type of physical process does this differential equation model?

    • Simple harmonic motion
    • Exponential growth
    • Constant acceleration
    • Exponential decay

  16. What is the integral of 1 / y with respect to y?

    • -1 / y^2
    • ln|y|
    • 1 / y^2
    • e^y

  17. In the equation dy/dx + y tan(x) = sec(x), what is P(x)?

    • tan(x)
    • cot(x)
    • sec(x)
    • 1

  18. What is lost in a general differential equation solution if you omit C?

    • Only the slope
    • Particular solution
    • The derivative
    • Family of curves

  19. Which differential equation cannot be solved using simple separation of variables?

    • dy/dx = x / y
    • dy/dx = xy
    • dy/dx = x + y
    • dy/dx = y / x

  20. What is the integrating factor for dy/dx + 3y = e^(2x)?

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

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