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″ + 3y′ + 2y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 3y′ + 2y = 0
r =
2.Easy
t (h)Nt = ?5000

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.17 h⁻¹. After how many hours t are there 5000 cells?

N0 = 500k = 0.17 h⁻¹N = 5000
t = h
3.Medium
t (min)T (°C)27Ta​T = ?

A cup of coffee at 94 °C is left in a room at 23 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.06 min⁻¹, find its temperature T after 27 min.

T0 = 94 °CTa = 23 °Ck = 0.06 min⁻¹t = 27 min
T = °C
4.Medium
t (min)T (°C)28Ta​23 °C13 °C

A can of soda at 6 °C is taken out of the fridge into a room at 27 °C. After 28 min its temperature is 23 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 13 °C?

T0 = 6 °CTa = 27 °CT(28) = 23 °CT = 13 °C
t = min
5.Hard
tyy(0) = 10y(0.25) = ?

Solve y″ + 2y′ + 5y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 2y′ + 5y = 0y(0) = 10y′(0) = 0
y(0.25) =
6.Hard
tyy(0) = 1y(1.5) = ?

Solve y″ + 3y′ + 27.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 3y′ + 27.25y = 0y(0) = 1y′(0) = 0
y(1.5) =

Answer key

  1. 1. r = −1
  2. 2. t = 13.54 h
  3. 3. T = 37.05 °C
  4. 4. t = 6.846 min
  5. 5. y(0.25) = 8.702
  6. 6. y(1.5) = 0.06619

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