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)y12 h425

A quantity decays according to dy/dt = −k·y with k > 0. It is 500 at t = 0 and 425 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 500y(12) = 425
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + 3y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3y′ + 36y = 0
ζ =
3.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 1 and y′(0) = 4. Find y(0.5).

y″ + 7y′ + 12y = 0y(0) = 1y′(0) = 4
y(0.5) =
4.Medium
t (min)T (°C)50Ta​103 °C

A forged steel part at 380 °C is left to cool in a workshop at 23 °C. After 50 min its temperature is 103 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 380 °CTa = 23 °CT(50) = 103 °C
k = min⁻¹
5.Hard
tyy(0) = 1y(1) = ?

Solve y″ + y′ + 9.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + y′ + 9.25y = 0y(0) = 1y′(0) = 0
y(1) =
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 3y′ + 6.25y = 6.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 6.25y = 6.25y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. k = 0.01354 h⁻¹
  2. 2. ζ = 0.25
  3. 3. y(0.5) = 0.8377
  4. 4. k = 0.02991 min⁻¹
  5. 5. y(1) = −0.5862
  6. 6. ymax = 1.095

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