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″ + 2.5y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.5y′ + 25y = 0
ζ =
2.Easy
t (h)y4 h975

A quantity grows according to dy/dt = k·y with k > 0. It is 250 at t = 0 and 975 at t = 4 h. Find the rate constant k in h⁻¹.

y(0) = 250y(4) = 975
k = h⁻¹
3.Medium
t (min)T (°C)16Ta​T = ?

A forged steel part at 320 °C is left to cool in a workshop at 29 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 16 min.

T0 = 320 °CTa = 29 °Ck = 0.05 min⁻¹t = 16 min
T = °C
4.Medium
t (min)T (°C)14Ta​40 °C

A cup of coffee at 84 °C is left in a room at 19 °C. After 14 min its temperature is 40 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 84 °CTa = 19 °CT(14) = 40 °C
k = min⁻¹
5.Hard
t (min)m (kg)25 minm = ?

A tank holds 1000 L of pure water. Brine with 0.1 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 25 min?

V = 1000 Lm0 = 0 kgc = 0.1 kg/Lr = 5 L/min
m = kg
6.Hard
tyy(0) = 3y(0.75) = ?

Solve y″ + 3y′ + 6.25y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 3y′ + 6.25y = 0y(0) = 3y′(0) = 0
y(0.75) =

Answer key

  1. 1. ζ = 0.25
  2. 2. k = 0.3402 h⁻¹
  3. 3. T = 159.8 °C
  4. 4. k = 0.0807 min⁻¹
  5. 5. m = 11.75 kg
  6. 6. y(0.75) = 0.7975

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