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″ + y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + y′ + 4y = 0
ζ =
2.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 =
3.Medium
t (min)T (°C)5Ta​73 °C

A cup of coffee at 82 °C is left in a room at 19 °C. After 5 min its temperature is 73 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 82 °CTa = 19 °CT(5) = 73 °C
k = min⁻¹
4.Medium
t (min)T (°C)30Ta​T = ?

A cup of coffee at 82 °C is left in a room at 20 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.07 min⁻¹, find its temperature T after 30 min.

T0 = 82 °CTa = 20 °Ck = 0.07 min⁻¹t = 30 min
T = °C
5.Hard
xy13(1, −3)y(3) = ?

Solve the initial value problem y′ + y/x = 3x2 for x > 0, with y(1) = −3. Find y(3).

y′ + y/x = 3x2y(1) = −3
y(3) =
6.Hard
t (min)m (kg)t = ?13 kg

A tank holds 100 L of water in which 20 kg of salt is dissolved. Brine with 0.1 kg of salt per litre flows in at 4 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 13 kg of salt?

V = 100 Lm0 = 20 kgc = 0.1 kg/Lr = 4 L/min
t = min

Answer key

  1. 1. ζ = 0.25
  2. 2. ζ = 0.1
  3. 3. k = 0.03083 min⁻¹
  4. 4. T = 27.59 °C
  5. 5. y(3) = 19
  6. 6. t = 30.1 min

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