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)y12 h460

A quantity grows according to dy/dt = k·y with k > 0. It is 200 at t = 0 and 460 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 200y(12) = 460
k = h⁻¹
2.Easy
t (h)Nt = ?1500

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.19 h⁻¹. After how many hours t are there 1500 cells?

N0 = 500k = 0.19 h⁻¹N = 1500
t = h
3.Medium
t (min)T (°C)14Ta​37 °C66 °C

A cup of coffee at 86 °C is left in a room at 21 °C. After 14 min its temperature is 37 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 66 °C?

T0 = 86 °CTa = 21 °CT(14) = 37 °CT = 66 °C
t = min
4.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 3 and y′(0) = 4. Find y(2).

y″ + 6y′ + 9y = 0y(0) = 3y′(0) = 4
y(2) =
5.Hard
tyK = 5y(1.5) = ?

A system obeys y″ + 3y′ + 11.25y = 56.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 3y′ + 11.25y = 56.25y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 2y′ + 17y = 85 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 17y = 85y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. k = 0.06941 h⁻¹
  2. 2. t = 5.782 h
  3. 3. t = 3.673 min
  4. 4. y(2) = 0.07188
  5. 5. y(1.5) = 5.369
  6. 6. ymax = 7.28

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