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
rr² + br + c?

For the equation y″ + 8y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 12y = 0
r =
2.Easy
t (h)Nt = ?3000

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.07 h⁻¹. After how many hours t are there 3000 cells?

N0 = 1000k = 0.07 h⁻¹N = 3000
t = h
3.Medium
t (min)T (°C)11Ta​57 °C80 °C

A cup of coffee at 95 °C is left in a room at 24 °C. After 11 min its temperature is 57 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 80 °C?

T0 = 95 °CTa = 24 °CT(11) = 57 °CT = 80 °C
t = min
4.Medium
tyy(0) = 3y(2) = ?

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

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = -2
y(2) =
5.Hard
tyy(0) = 2y(1.5) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 2y′(0) = 0
y(1.5) =
6.Hard
ty2y(2) = ?

Solve y′ + 2y = 9e−3t with y(0) = 3. Find y(2).

y′ + 2y = 9e−3ty(0) = 3
y(2) =

Answer key

  1. 1. r = −2
  2. 2. t = 15.69 h
  3. 3. t = 3.407 min
  4. 4. y(2) = 0.2015
  5. 5. y(1.5) = 0.2384
  6. 6. y(2) = 0.1975

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