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
t (h)y6 h30

A quantity decays according to dy/dt = −k·y with k > 0. It is 120 at t = 0 and 30 at t = 6 h. Find the rate constant k in h⁻¹.

y(0) = 120y(6) = 30
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + 1.8y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.8y′ + 9y = 0
ζ =
3.Medium
t (min)T (°C)22Ta​44 °C31 °C

A cup of coffee at 85 °C is left in a room at 24 °C. After 22 min its temperature is 44 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 31 °C?

T0 = 85 °CTa = 24 °CT(22) = 44 °CT = 31 °C
t = min
4.Medium
tyy(0) = 4y(2) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 4 and y′(0) = -4. Find y(2).

y″ + 7y′ + 12y = 0y(0) = 4y′(0) = -4
y(2) =
5.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 3y′ + 18.25y = 36.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 18.25y = 36.5y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 10y(1.5) = ?

A system obeys y″ + 3y′ + 18.25y = 182.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 3y′ + 18.25y = 182.5y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. k = 0.231 h⁻¹
  2. 2. ζ = 0.3
  3. 3. t = 42.71 min
  4. 4. y(2) = 0.02706
  5. 5. ymax = 2.616
  6. 6. y(1.5) = 9.098

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