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 (days)m (mg)48 dm = ?

A radioactive tracer has a half-life of 13 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 48 days?

m0 = 120 mgt½ = 13 dt = 48 d
m = mg
2.Easy
ty1

A mass–spring–damper is described by y″ + 2.4y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 4y = 0
ζ =
3.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 3 and y′(0) = -1. Find y(2).

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = -1
y(2) =
4.Medium
t (min)T (°C)25Ta​T = ?

A cup of coffee at 90 °C is left in a room at 21 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.1 min⁻¹, find its temperature T after 25 min.

T0 = 90 °CTa = 21 °Ck = 0.1 min⁻¹t = 25 min
T = °C
5.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 4y′ + 20y = 40 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 20y = 40y(0) = 0y′(0) = 0
ymax =
6.Hard
ty1y(1) = ?

Solve y′ + 2y = 10e−3t with y(0) = 3. Find y(1).

y′ + 2y = 10e−3ty(0) = 3
y(1) =

Answer key

  1. 1. m = 9.283 mg
  2. 2. ζ = 0.6
  3. 3. y(2) = 0.2381
  4. 4. T = 26.66 °C
  5. 5. ymax = 2.416
  6. 6. y(1) = 1.261

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