Differential equations worksheet with answers

Free printable Differential equations worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
ty1

A mass–spring–damper is described by y″ + 6y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6y′ + 4y = 0
ζ =
2.Easy
t (h)Nt = ?4000

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.09 h⁻¹. After how many hours t are there 4000 cells?

N0 = 2000k = 0.09 h⁻¹N = 4000
t = h
3.Medium
t (min)T (°C)5Ta​72 °C52 °C

A cup of coffee at 89 °C is left in a room at 22 °C. After 5 min its temperature is 72 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 52 °C?

T0 = 89 °CTa = 22 °CT(5) = 72 °CT = 52 °C
t = min
4.Medium
t (min)T (°C)6Ta​66 °C78 °C

A cup of coffee at 86 °C is left in a room at 20 °C. After 6 min its temperature is 66 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 78 °C?

T0 = 86 °CTa = 20 °CT(6) = 66 °CT = 78 °C
t = min
5.Hard
t (min)m (kg)t = ?45 kg

A tank holds 500 L of water in which 10 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 45 kg of salt?

V = 500 Lm0 = 10 kgc = 0.25 kg/Lr = 5 L/min
t = min
6.Hard
t (min)m (kg)5 minm = ?

A tank holds 100 L of pure water. Brine with 0.05 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 5 min?

V = 100 Lm0 = 0 kgc = 0.05 kg/Lr = 10 L/min
m = kg

Answer key

  1. 1. ζ = 1.5
  2. 2. t = 7.702 h
  3. 3. t = 13.73 min
  4. 4. t = 2.147 min
  5. 5. t = 36.29 min
  6. 6. m = 1.967 kg

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