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″ + 24y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 24y′ + 100y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 0.8y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 0.8y′ + 16y = 0
ζ =
3.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 2 and y′(0) = 6. Find y(0.5).

y″ + 8y′ + 15y = 0y(0) = 2y′(0) = 6
y(0.5) =
4.Medium
t (min)T (°C)29Ta​34 °C59 °C

A cup of coffee at 84 °C is left in a room at 19 °C. After 29 min its temperature is 34 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 59 °C?

T0 = 84 °CTa = 19 °CT(29) = 34 °CT = 59 °C
t = min
5.Hard
xy12(1, 3)y(2) = ?

Solve the initial value problem y′ + 3y/x = 6 for x > 0, with y(1) = 3. Find y(2).

y′ + 3y/x = 6y(1) = 3
y(2) =
6.Hard
tyy(0) = 3y(0.75) = ?

Solve y″ + 4y′ + 8y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 3y′(0) = 0
y(0.75) =

Answer key

  1. 1. ζ = 1.2
  2. 2. ζ = 0.1
  3. 3. y(0.5) = 1.293
  4. 4. t = 9.602 min
  5. 5. y(2) = 3.188
  6. 6. y(0.75) = 0.7151

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