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
rr² + br + c?

For the equation y″ + 7y′ + 6y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 6y = 0
r =
3.Medium
t (min)T (°C)6Ta​13 °C

A can of soda at 4 °C is taken out of the fridge into a room at 25 °C. After 6 min its temperature is 13 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 4 °CTa = 25 °CT(6) = 13 °C
k = min⁻¹
4.Medium
tyy(0) = 2y(2) = ?

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

y″ + 8y′ + 16y = 0y(0) = 2y′(0) = 1
y(2) =
5.Hard
tyK = 5y(0.75) = ?

A system obeys y″ + 4y′ + 29y = 145 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 4y′ + 29y = 145y(0) = 0y′(0) = 0
y(0.75) =
6.Hard
ty1y(1) = ?

Solve y′ + 2y = 4e−3t with y(0) = 3. Find y(1).

y′ + 2y = 4e−3ty(0) = 3
y(1) =

Answer key

  1. 1. r = −2
  2. 2. r = −1
  3. 3. k = 0.09327 min⁻¹
  4. 4. y(2) = 0.006709
  5. 5. y(0.75) = 6.171
  6. 6. y(1) = 0.7482

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