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″ + 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 =
2.Easy
t (h)N2 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.37 h⁻¹. How many cells N are there after 2 h?

N0 = 500k = 0.37 h⁻¹t = 2 h
N =
3.Medium
t (min)T (°C)20Ta​33 °C56 °C

A cup of coffee at 90 °C is left in a room at 19 °C. After 20 min its temperature is 33 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 56 °C?

T0 = 90 °CTa = 19 °CT(20) = 33 °CT = 56 °C
t = min
4.Medium
t (min)T (°C)14Ta​14 °C

A can of soda at 6 °C is taken out of the fridge into a room at 25 °C. After 14 min its temperature is 14 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 6 °CTa = 25 °CT(14) = 14 °C
k = min⁻¹
5.Hard
ty2y(2) = ?

Solve y′ + y = 6e−2t with y(0) = 0. Find y(2).

y′ + y = 6e−2ty(0) = 0
y(2) =
6.Hard
tyy(0) = 5y(0.5) = ?

Solve y″ + 3y′ + 27.25y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 3y′ + 27.25y = 0y(0) = 5y′(0) = 0
y(0.5) =

Answer key

  1. 1. r = −1
  2. 2. N = 1048
  3. 3. t = 8.028 min
  4. 4. k = 0.03904 min⁻¹
  5. 5. y(2) = 0.7021
  6. 6. y(0.5) = −1.468

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