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″ + 2y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2y′ + 16y = 0
ζ =
2.Easy
t (h)Nt = ?10000

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.39 h⁻¹. After how many hours t are there 10000 cells?

N0 = 5000k = 0.39 h⁻¹N = 10000
t = h
3.Medium
t (min)T (°C)22Ta​T = ?

A forged steel part at 480 °C is left to cool in a workshop at 23 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 22 min.

T0 = 480 °CTa = 23 °Ck = 0.04 min⁻¹t = 22 min
T = °C
4.Medium
t (min)T (°C)52Ta​52 °C37 °C

A forged steel part at 440 °C is left to cool in a workshop at 21 °C. After 52 min its temperature is 52 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 37 °C?

T0 = 440 °CTa = 21 °CT(52) = 52 °CT = 37 °C
t = min
5.Hard
t (min)m (kg)t = ?80 kg

A tank holds 1000 L of pure water. Brine with 0.1 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 = 0.1 kg/Lr = 2 L/min
t = min
6.Hard
ty1.5y(1.5) = ?

Solve y′ + 3y = 7e−t with y(0) = 1. Find y(1.5).

y′ + 3y = 7e−ty(0) = 1
y(1.5) =

Answer key

  1. 1. ζ = 0.25
  2. 2. t = 1.777 h
  3. 3. T = 212.6 °C
  4. 4. t = 65.21 min
  5. 5. t = 804.7 min
  6. 6. y(1.5) = 0.7532

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