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 (h)Nt = ?5000

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.36 h⁻¹. After how many hours t are there 5000 cells?

N0 = 500k = 0.36 h⁻¹N = 5000
t = h
2.Easy
t (h)y4 h54

A quantity decays according to dy/dt = −k·y with k > 0. It is 120 at t = 0 and 54 at t = 4 h. Find the rate constant k in h⁻¹.

y(0) = 120y(4) = 54
k = h⁻¹
3.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 5 and y′(0) = -2. Find y(0.5).

y″ + 4y′ + 4y = 0y(0) = 5y′(0) = -2
y(0.5) =
4.Medium
tyy(0) = 4y(2) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 4 and y′(0) = -4. Find y(2).

y″ + 8y′ + 15y = 0y(0) = 4y′(0) = -4
y(2) =
5.Hard
tyy(0) = 5y(0.25) = ?

Solve y″ + y′ + 16.25y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + y′ + 16.25y = 0y(0) = 5y′(0) = 0
y(0.25) =
6.Hard
tyy(0) = 10y(0.5) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 10y′(0) = 0
y(0.5) =

Answer key

  1. 1. t = 6.396 h
  2. 2. k = 0.1996 h⁻¹
  3. 3. y(0.5) = 3.311
  4. 4. y(2) = 0.01965
  5. 5. y(0.25) = 2.848
  6. 6. y(0.5) = −4.133

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