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
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)23Ta​26 °C

A cup of coffee at 90 °C is left in a room at 19 °C. After 23 min its temperature is 26 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 90 °CTa = 19 °CT(23) = 26 °C
k = min⁻¹
4.Medium
t (min)T (°C)48Ta​54 °C

A forged steel part at 360 °C is left to cool in a workshop at 23 °C. After 48 min its temperature is 54 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 360 °CTa = 23 °CT(48) = 54 °C
k = min⁻¹
5.Hard
t (min)m (kg)40 minm = ?

A tank holds 400 L of water in which 5 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 20 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 40 min?

V = 400 Lm0 = 5 kgc = 0.25 kg/Lr = 20 L/min
m = kg
6.Hard
ty2y(2) = ?

Solve y′ + 3y = 9e−2t with y(0) = 3. Find y(2).

y′ + 3y = 9e−2ty(0) = 3
y(2) =

Answer key

  1. 1. ζ = 0.25
  2. 2. ζ = 0.6
  3. 3. k = 0.1007 min⁻¹
  4. 4. k = 0.04971 min⁻¹
  5. 5. m = 87.14 kg
  6. 6. y(2) = 0.15

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