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″ + 1.2y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.2y′ + 9y = 0
ζ =
2.Easy
t (h)y10 h336

A quantity grows according to dy/dt = k·y with k > 0. It is 120 at t = 0 and 336 at t = 10 h. Find the rate constant k in h⁻¹.

y(0) = 120y(10) = 336
k = h⁻¹
3.Medium
t (min)T (°C)31Ta​115 °C

A forged steel part at 340 °C is left to cool in a workshop at 24 °C. After 31 min its temperature is 115 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 340 °CTa = 24 °CT(31) = 115 °C
k = min⁻¹
4.Medium
tyy(0) = 2y(2) = ?

Solve y″ + 6y′ + 5y = 0 with y(0) = 2 and y′(0) = 2. Find y(2).

y″ + 6y′ + 5y = 0y(0) = 2y′(0) = 2
y(2) =
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + y′ + 16.25y = 16.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 16.25y = 16.25y(0) = 0y′(0) = 0
ymax =
6.Hard
xy13(1, −3)y(3) = ?

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

y′ + y/x = 1y(1) = −3
y(3) =

Answer key

  1. 1. ζ = 0.2
  2. 2. k = 0.103 h⁻¹
  3. 3. k = 0.04016 min⁻¹
  4. 4. y(2) = 0.406
  5. 5. ymax = 1.675
  6. 6. y(3) = 0.3333

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