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″ + 7.2y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 7.2y′ + 9y = 0
ζ =
2.Easy
t (days)m (mg)99 dm = ?

A radioactive tracer has a half-life of 29 days. A sample initially contains 80 mg. The decay follows dm/dt = −k·m. How much m remains after 99 days?

m0 = 80 mgt½ = 29 dt = 99 d
m = mg
3.Medium
t (min)T (°C)13Ta​237 °C

A forged steel part at 380 °C is left to cool in a workshop at 28 °C. After 13 min its temperature is 237 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 380 °CTa = 28 °CT(13) = 237 °C
k = min⁻¹
4.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 6y′ + 5y = 0 with y(0) = 5 and y′(0) = -2. Find y(0.5).

y″ + 6y′ + 5y = 0y(0) = 5y′(0) = -2
y(0.5) =
5.Hard
tyK = 2ymax​ = ?

A system obeys y″ + y′ + 4.25y = 8.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 4.25y = 8.5y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 3y′ + 27.25y = 136.3 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 27.25y = 136.3y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 1.2
  2. 2. m = 7.506 mg
  3. 3. k = 0.0401 min⁻¹
  4. 4. y(0.5) = 3.426
  5. 5. ymax = 2.912
  6. 6. ymax = 6.948

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