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
rr² + br + c?

For the equation y″ + 4y′ + 3y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 4y′ + 3y = 0
r =
2.Easy
ty1

A mass–spring–damper is described by y″ + 12y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 12y′ + 100y = 0
ζ =
3.Medium
t (min)T (°C)51Ta​123 °C230 °C

A forged steel part at 480 °C is left to cool in a workshop at 24 °C. After 51 min its temperature is 123 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 230 °C?

T0 = 480 °CTa = 24 °CT(51) = 123 °CT = 230 °C
t = min
4.Medium
t (min)T (°C)8Ta​T = ?

A cup of coffee at 87 °C is left in a room at 23 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.08 min⁻¹, find its temperature T after 8 min.

T0 = 87 °CTa = 23 °Ck = 0.08 min⁻¹t = 8 min
T = °C
5.Hard
t (min)m (kg)t = ?10 kg

A tank holds 400 L of water in which 5 kg of salt is dissolved. Brine with 0.05 kg of salt per litre flows in at 20 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 10 kg of salt?

V = 400 Lm0 = 5 kgc = 0.05 kg/Lr = 20 L/min
t = min
6.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 2y′ + 10y = 50 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 10y = 50y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. r = −1
  2. 2. ζ = 0.6
  3. 3. t = 26.53 min
  4. 4. T = 56.75 °C
  5. 5. t = 8.109 min
  6. 6. ymax = 6.755

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