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″ + 1.6y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.6y′ + 64y = 0
ζ =
2.Easy
t (h)y10 h300

A quantity decays according to dy/dt = −k·y with k > 0. It is 400 at t = 0 and 300 at t = 10 h. Find the rate constant k in h⁻¹.

y(0) = 400y(10) = 300
k = h⁻¹
3.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 3 and y′(0) = -2. Find y(1.5).

y″ + 7y′ + 10y = 0y(0) = 3y′(0) = -2
y(1.5) =
4.Medium
t (min)T (°C)15Ta​192 °C

A forged steel part at 380 °C is left to cool in a workshop at 23 °C. After 15 min its temperature is 192 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 380 °CTa = 23 °CT(15) = 192 °C
k = min⁻¹
5.Hard
tyK = 2y(1.5) = ?

A system obeys y″ + y′ + 16.25y = 32.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + y′ + 16.25y = 32.5y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
t (min)m (kg)t = ?28 kg

A tank holds 400 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. After how many minutes t does the tank contain 28 kg of salt?

V = 400 Lm0 = 0 kgc = 0.1 kg/Lr = 5 L/min
t = min

Answer key

  1. 1. ζ = 0.1
  2. 2. k = 0.02877 h⁻¹
  3. 3. y(1.5) = 0.215
  4. 4. k = 0.04986 min⁻¹
  5. 5. y(1.5) = 1.126
  6. 6. t = 96.32 min

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