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
t (h)y10 h55

A quantity decays according to dy/dt = −k·y with k > 0. It is 100 at t = 0 and 55 at t = 10 h. Find the rate constant k in h⁻¹.

y(0) = 100y(10) = 55
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + 19.2y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 19.2y′ + 64y = 0
ζ =
3.Medium
t (min)T (°C)6Ta​10 °C

A can of soda at 5 °C is taken out of the fridge into a room at 27 °C. After 6 min its temperature is 10 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 5 °CTa = 27 °CT(6) = 10 °C
k = min⁻¹
4.Medium
t (min)T (°C)22Ta​T = ?

A can of soda at 4 °C is taken out of the fridge into a room at 23 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.07 min⁻¹, find its temperature T after 22 min.

T0 = 4 °CTa = 23 °Ck = 0.07 min⁻¹t = 22 min
T = °C
5.Hard
tyy(0) = 10y(0.25) = ?

Solve y″ + 2y′ + 10y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 2y′ + 10y = 0y(0) = 10y′(0) = 0
y(0.25) =
6.Hard
tyy(0) = 3y(1.5) = ?

Solve y″ + 4y′ + 20y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 4y′ + 20y = 0y(0) = 3y′(0) = 0
y(1.5) =

Answer key

  1. 1. k = 0.05978 h⁻¹
  2. 2. ζ = 1.2
  3. 3. k = 0.04297 min⁻¹
  4. 4. T = 18.93 °C
  5. 5. y(0.25) = 7.468
  6. 6. y(1.5) = 0.1225

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