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″ + 0.6y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 0.6y′ + 9y = 0
ζ =
2.Easy
t (h)Nt = ?400

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.05 h⁻¹. After how many hours t are there 400 cells?

N0 = 200k = 0.05 h⁻¹N = 400
t = h
3.Medium
t (min)T (°C)11Ta​17 °C

A can of soda at 6 °C is taken out of the fridge into a room at 22 °C. After 11 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 = 6 °CTa = 22 °CT(11) = 17 °C
k = min⁻¹
4.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 2 and y′(0) = -4. Find y(1.5).

y″ + 7y′ + 12y = 0y(0) = 2y′(0) = -4
y(1.5) =
5.Hard
t (min)m (kg)t = ?7 kg

A tank holds 100 L of water in which 5 kg of salt is dissolved. Brine with 0.1 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 7 kg of salt?

V = 100 Lm0 = 5 kgc = 0.1 kg/Lr = 10 L/min
t = min
6.Hard
tyy(0) = 2y(0.75) = ?

Solve y″ + 4y′ + 20y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 4y′ + 20y = 0y(0) = 2y′(0) = 0
y(0.75) =

Answer key

  1. 1. ζ = 0.1
  2. 2. t = 13.86 h
  3. 3. k = 0.1057 min⁻¹
  4. 4. y(1.5) = 0.03948
  5. 5. t = 5.108 min
  6. 6. y(0.75) = −0.4103

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