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Differential equations: board exam questions with solutions

On the ECE exam under Mathematics and the REE exam under Mathematics. Try each question before you open its solution, then practise with fresh numbers below: 2 computed question types on this topic.

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Questions

1. What are the order and degree of (d²y/dx²)³ + (dy/dx)⁴ = x?

  1. Order 3, degree 2
  2. Order 2, degree 3
  3. Order 2, degree 4
  4. Order 4, degree 2
Show the answer and solution

Answer: B. Order 2, degree 3

The highest derivative is the second, so the order is 2. That derivative is raised to the power 3, so the degree is 3.

2. Solve dy/dx = 2y with y(0) = 5, then find y(1).

  1. 10.00
  2. 36.95
  3. 13.59
  4. 73.89
Show the answer and solution

Answer: B. 36.95

Separating variables gives y = 5e^(2x), so y(1) = 5e² = 36.95.

3. The characteristic roots of y″ + 4y′ + 13y = 0 are:

  1. 2 ± j3
  2. −4 ± j13
  3. −2 ± j3
  4. −1 and −13
Show the answer and solution

Answer: C. −2 ± j3

r² + 4r + 13 = 0, so r = [−4 ± √(16 − 52)]/2 = −2 ± j3. The response is a decaying oscillation.

4. A 100 µF capacitor charges from a 12 V DC source through 10 kΩ, starting uncharged. Find the capacitor voltage at t = 2 s.

  1. 7.59 V
  2. 1.62 V
  3. 12.00 V
  4. 10.38 V
Show the answer and solution

Answer: D. 10.38 V

τ = RC = 10 000 × 100×10⁻⁶ = 1 s, so v = 12(1 − e^(−2)) = 12(0.8647) = 10.38 V.

5. What is the integrating factor of dy/dx + 3y = x?

  1. e^(3x)
  2. e^(−3x)
  3. e^(3x²/2)
  4. 3x
Show the answer and solution

Answer: A. e^(3x)

For y′ + P(x)y = Q(x), the integrating factor is e^(∫P dx). Here P = 3, so it is e^(3x).

6. Assume Newton's law of cooling holds. A body cools from 90 °C to 70 °C in 10 min in a room at 30 °C. What is its temperature after 20 min?

  1. 50.00 °C
  2. 60.00 °C
  3. 46.67 °C
  4. 56.67 °C
Show the answer and solution

Answer: D. 56.67 °C

T − 30 = 60e^(−kt). At 10 min, e^(−10k) = 40/60 = 2/3. At 20 min, T = 30 + 60(2/3)² = 30 + 26.67 = 56.67 °C. 50 °C would be a linear drop.

7. Which of these differential equations is linear?

  1. y′ + y² = x
  2. y·y′ = 1
  3. y′ + x²y = sin x
  4. y′ = sin y
Show the answer and solution

Answer: C. y′ + x²y = sin x

In a linear ODE, y and its derivatives appear only to the first power and are not multiplied together or placed inside functions. Only y′ + x²y = sin x qualifies.

8. Solve y″ + 5y′ + 6y = 0 with y(0) = 1 and y′(0) = 0.

  1. y = 3e^(−2x) − 2e^(−3x)
  2. y = 2e^(−2x) − e^(−3x)
  3. y = −2e^(−2x) + 3e^(−3x)
  4. y = e^(−2x)
Show the answer and solution

Answer: A. y = 3e^(−2x) − 2e^(−3x)

The roots are −2 and −3, so y = Ae^(−2x) + Be^(−3x). The initial conditions give A + B = 1 and −2A − 3B = 0, so A = 3 and B = −2.

Worked examples with computed numbers

These numbers are fixed on this page; in practice they change every time.

9. A substance decays at a rate proportional to the amount present, with a half-life of 24 years. How many years until 10% of it remains?

  1. 55.3
  2. 79.7
  3. 43.2
  4. 120
Show the answer and solution

Answer: B. 79.7

dN/dt = −kN gives N = N₀e^(−kt) with k = ln2/24; t = ln(1/0.1)/k = 79.7 years.

10. Water at 83 °C cools to 61 °C in 12 minutes in a 30 °C room. By Newton's law of cooling, what is its temperature after 24 minutes?

  1. 45.5 °C
  2. 40.3 °C
  3. 39 °C
  4. 48.1 °C
Show the answer and solution

Answer: D. 48.1 °C

T − 30 = 53e^(−kt); from the first reading k = ln(53/31)/12 = 0.0447/min, so at 24 min T = 48.1 °C.

Other topics in the same subjects

Algebra · Trigonometry · Analytic geometry · Differential calculus · Integral calculus · Complex numbers · Probability and statistics · Matrices and determinants · Laplace transforms and Fourier series

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