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″ + 12y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 12y′ + 100y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 5y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 5y′ + 25y = 0
ζ =
3.Medium
t (min)T (°C)13Ta​61 °C

A cup of coffee at 80 °C is left in a room at 22 °C. After 13 min its temperature is 61 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 80 °CTa = 22 °CT(13) = 61 °C
k = min⁻¹
4.Medium
t (min)T (°C)13Ta​T = ?

A can of soda at 6 °C is taken out of the fridge into a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.08 min⁻¹, find its temperature T after 13 min.

T0 = 6 °CTa = 22 °Ck = 0.08 min⁻¹t = 13 min
T = °C
5.Hard
t (min)m (kg)25 minm = ?

A tank holds 100 L of pure water. Brine with 0.25 kg of salt per litre flows in at 4 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 = 100 Lm0 = 0 kgc = 0.25 kg/Lr = 4 L/min
m = kg
6.Hard
xy13(1, −2)y(3) = ?

Solve the initial value problem y′ + y/x = 2x for x > 0, with y(1) = −2. Find y(3).

y′ + y/x = 2xy(1) = −2
y(3) =

Answer key

  1. 1. ζ = 0.6
  2. 2. ζ = 0.5
  3. 3. k = 0.03053 min⁻¹
  4. 4. T = 16.34 °C
  5. 5. m = 15.8 kg
  6. 6. y(3) = 5.111

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