Differential Equations Intro Quiz

Grade 12 · Math · Quiz

10 questions on separable DEs, slope fields, and exponential growth/decay models.

Questions

  1. What is the order of the differential equation d²y/dx² + 3y = 0?

    1. First order
    2. Second order
    3. Third order
    4. Zero order
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    Answer: Second order

    The order equals the highest derivative present. Here that is d²y/dx², which is second order.

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

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    y dy = x dx → y²/2 = x²/2 + C → y² = x² + K. y(0)=3: 9 = 0 + K → y² = x² + 9, so y = sqrt(x²+9).

  3. In a slope field for dy/dx = x, the slope at (2, 5) is:

    1. 5
    2. 2
    3. 10
    4. 0
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    Answer: 2

    dy/dx = x, so at x = 2 the slope is 2, regardless of y.

  4. The equation dy/dx = x + y is separable.

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    Separable means dy/dx = f(x)·g(y). x + y cannot be factored into separate x and y functions.

  5. A radioactive substance decays with dA/dt = -0.1A. Starting with A(0) = 50, what is A(t)?

    1. 50e^(0.1t)
    2. 50e^(-0.1t)
    3. 50 - 0.1t
    4. -0.1(50)
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    Answer: 50e^(-0.1t)

    Exponential decay: A = A₀ e^(kt). A₀ = 50, k = -0.1. A(t) = 50e^(-0.1t).

  6. Verify that y = e^(3x) is a solution to dy/dx = 3y.

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    dy/dx of e^(3x) = 3e^(3x). And 3y = 3e^(3x). Both sides are equal, so it is a solution.

  7. The general solution of dy/dx = 4 is:

    1. y = 4x
    2. y = 4x + C
    3. y = 2x² + C
    4. y = C
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    Answer: y = 4x + C

    dy = 4 dx → y = 4x + C.

  8. A particular solution to a DE is unique once an initial condition is specified.

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    Under standard smoothness conditions, an initial condition determines a unique solution (Existence and Uniqueness Theorem).

  9. Solve dy/dx = e^x with y(0) = 2.

    1. y = eˣ
    2. y = eˣ + 1
    3. y = eˣ + 2
    4. y = 2eˣ
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    Answer: y = eˣ + 1

    y = ∫ eˣ dx = eˣ + C. y(0)=2: 2 = e⁰ + C = 1 + C → C = 1. So y = eˣ + 1.

  10. A quantity grows at 5% per year. Write the differential equation and its solution if the initial amount is Q₀.

    Show answer

    Proportional growth → dQ/dt = kQ with k = 0.05. Solution: Q = Q₀ e^(0.05t).

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