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″ + 4y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4y′ + 64y = 0
ζ =
2.Easy
t (days)m (mg)36 dm = ?

A radioactive tracer has a half-life of 21 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 36 days?

m0 = 10 mgt½ = 21 dt = 36 d
m = mg
3.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 5 and y′(0) = -1. Find y(2).

y″ + 4y′ + 4y = 0y(0) = 5y′(0) = -1
y(2) =
4.Medium
t (min)T (°C)35Ta​T = ?

A forged steel part at 440 °C is left to cool in a workshop at 25 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 35 min.

T0 = 440 °CTa = 25 °Ck = 0.05 min⁻¹t = 35 min
T = °C
5.Hard
tyy(0) = 5y(0.5) = ?

Solve y″ + 4y′ + 29y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 4y′ + 29y = 0y(0) = 5y′(0) = 0
y(0.5) =
6.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 4y′ + 20y = 100 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 20y = 100y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.25
  2. 2. m = 3.048 mg
  3. 3. y(2) = 0.4213
  4. 4. T = 97.12 °C
  5. 5. y(0.5) = −1.033
  6. 6. ymax = 6.039

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