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″ + 5y′ + 4y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 4y = 0
r =
2.Easy
ty1

A mass–spring–damper is described by y″ + 3.2y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3.2y′ + 16y = 0
ζ =
3.Medium
t (min)T (°C)36Ta​123 °C

A forged steel part at 440 °C is left to cool in a workshop at 24 °C. After 36 min its temperature is 123 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 440 °CTa = 24 °CT(36) = 123 °C
k = min⁻¹
4.Medium
t (min)T (°C)30Ta​28 °C

A cup of coffee at 82 °C is left in a room at 24 °C. After 30 min its temperature is 28 °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(30) = 28 °C
k = min⁻¹
5.Hard
tyy(0) = 1y(0.25) = ?

Solve y″ + 3y′ + 18.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 3y′ + 18.25y = 0y(0) = 1y′(0) = 0
y(0.25) =
6.Hard
t (min)m (kg)t = ?80 kg

A tank holds 500 L of pure water. Brine with 0.2 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 80 kg of salt?

V = 500 Lm0 = 0 kgc = 0.2 kg/Lr = 5 L/min
t = min

Answer key

  1. 1. r = −1
  2. 2. ζ = 0.4
  3. 3. k = 0.03988 min⁻¹
  4. 4. k = 0.08914 min⁻¹
  5. 5. y(0.25) = 0.5882
  6. 6. t = 160.9 min

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