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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
ty1

A mass–spring–damper is described by y″ + 2y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2y′ + 4y = 0
ζ =
2.Easy
t (h)N9 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.21 h⁻¹. How many cells N are there after 9 h?

N0 = 5000k = 0.21 h⁻¹t = 9 h
N =
3.Medium
t (min)T (°C)15Ta​19 °C

A can of soda at 5 °C is taken out of the fridge into a room at 24 °C. After 15 min its temperature is 19 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 5 °CTa = 24 °CT(15) = 19 °C
k = min⁻¹
4.Medium
t (min)T (°C)13Ta​13 °C8 °C

A can of soda at 3 °C is taken out of the fridge into a room at 22 °C. After 13 min its temperature is 13 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 8 °C?

T0 = 3 °CTa = 22 °CT(13) = 13 °CT = 8 °C
t = min
5.Hard
tyy(0) = 2y(0.25) = ?

Solve y″ + y′ + 25.25y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + y′ + 25.25y = 0y(0) = 2y′(0) = 0
y(0.25) =
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + y′ + 9.25y = 9.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 9.25y = 9.25y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.5
  2. 2. N = 33097
  3. 3. k = 0.089 min⁻¹
  4. 4. t = 5.313 min
  5. 5. y(0.25) = 0.724
  6. 6. ymax = 1.592

Full step-by-step solutions: practise this topic on examproblems.com.