1.Easy
For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.
y″ + 7y′ + 12y = 0
r =
Differential equations worksheet with answers
2.Easy
A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.11 h⁻¹. How many cells N are there after 20 h?
N0 = 500k = 0.11 h⁻¹t = 20 h
N =
3.Medium
A forged steel part at 460 °C is left to cool in a workshop at 28 °C. After 30 min its temperature is 265 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 343 °C?
T0 = 460 °CTa = 28 °CT(30) = 265 °CT = 343 °C
t = min
4.Medium
A can of soda at 5 °C is taken out of the fridge into a room at 27 °C. After 10 min its temperature is 17 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.
T0 = 5 °CTa = 27 °CT(10) = 17 °C
k = min⁻¹
5.Hard
A tank holds 1000 L of pure water. Brine with 0.2 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 80 kg of salt?
V = 1000 Lm0 = 0 kgc = 0.2 kg/Lr = 2 L/min
t = min
6.Hard
A system obeys y″ + 4y′ + 13y = 65 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.
y″ + 4y′ + 13y = 65y(0) = 0y′(0) = 0
ymax =
Answer key
- 1. r = −3
- 2. N = 4513
- 3. t = 15.78 min
- 4. k = 0.07885 min⁻¹
- 5. t = 255.4 min
- 6. ymax = 5.616
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