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″ + 8y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 8y′ + 64y = 0
ζ =
2.Easy
t (h)y2 h200

A quantity grows according to dy/dt = k·y with k > 0. It is 50 at t = 0 and 200 at t = 2 h. Find the rate constant k in h⁻¹.

y(0) = 50y(2) = 200
k = h⁻¹
3.Medium
t (min)T (°C)11Ta​51 °C

A cup of coffee at 82 °C is left in a room at 24 °C. After 11 min its temperature is 51 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 82 °CTa = 24 °CT(11) = 51 °C
k = min⁻¹
4.Medium
t (min)T (°C)25Ta​T = ?

A cup of coffee at 86 °C is left in a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 25 min.

T0 = 86 °CTa = 22 °Ck = 0.03 min⁻¹t = 25 min
T = °C
5.Hard
tyy(0) = 5y(0.5) = ?

Solve y″ + y′ + 4.25y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + y′ + 4.25y = 0y(0) = 5y′(0) = 0
y(0.5) =
6.Hard
t (min)m (kg)t = ?83 kg

A tank holds 500 L of water in which 20 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 83 kg of salt?

V = 500 Lm0 = 20 kgc = 0.25 kg/Lr = 5 L/min
t = min

Answer key

  1. 1. ζ = 0.5
  2. 2. k = 0.6931 h⁻¹
  3. 3. k = 0.06951 min⁻¹
  4. 4. T = 52.23 °C
  5. 5. y(0.5) = 2.923
  6. 6. t = 91.63 min

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