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)99 dm = ?

A radioactive tracer has a half-life of 29 days. A sample initially contains 100 mg. The decay follows dm/dt = −k·m. How much m remains after 99 days?

m0 = 100 mgt½ = 29 dt = 99 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 12y = 0
r =
3.Medium
t (min)T (°C)11Ta​56 °C70 °C

A cup of coffee at 83 °C is left in a room at 19 °C. After 11 min its temperature is 56 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 70 °C?

T0 = 83 °CTa = 19 °CT(11) = 56 °CT = 70 °C
t = min
4.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 5 and y′(0) = -2. Find y(1).

y″ + 8y′ + 16y = 0y(0) = 5y′(0) = -2
y(1) =
5.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 4y′ + 29y = 58 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 29y = 58y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 2y′ + 17y = 17 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 17y = 17y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. m = 9.383 mg
  2. 2. r = −3
  3. 3. t = 4.558 min
  4. 4. y(1) = 0.4213
  5. 5. ymax = 2.569
  6. 6. ymax = 1.456

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