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)y11 h1560

A quantity grows according to dy/dt = k·y with k > 0. It is 400 at t = 0 and 1560 at t = 11 h. Find the rate constant k in h⁻¹.

y(0) = 400y(11) = 1560
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + 6y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6y′ + 25y = 0
ζ =
3.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 5 and y′(0) = 4. Find y(1).

y″ + 4y′ + 4y = 0y(0) = 5y′(0) = 4
y(1) =
4.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 1 and y′(0) = 6. Find y(1.5).

y″ + 7y′ + 12y = 0y(0) = 1y′(0) = 6
y(1.5) =
5.Hard
ty0.5y(0.5) = ?

Solve y′ + y = 4e−3t with y(0) = 5. Find y(0.5).

y′ + y = 4e−3ty(0) = 5
y(0.5) =
6.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 4y′ + 29y = 145 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 29y = 145y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. k = 0.1237 h⁻¹
  2. 2. ζ = 0.6
  3. 3. y(1) = 2.571
  4. 4. y(1.5) = 0.08878
  5. 5. y(0.5) = 3.799
  6. 6. ymax = 6.423

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