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)N24 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.03 h⁻¹. How many cells N are there after 24 h?

N0 = 5000k = 0.03 h⁻¹t = 24 h
N =
2.Easy
rr² + br + c?

For the equation y″ + 6y′ + 8y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 8y = 0
r =
3.Medium
t (min)T (°C)22Ta​45 °C

A cup of coffee at 90 °C is left in a room at 23 °C. After 22 min its temperature is 45 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 90 °CTa = 23 °CT(22) = 45 °C
k = min⁻¹
4.Medium
tyy(0) = 3y(1) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 3 and y′(0) = 1. Find y(1).

y″ + 7y′ + 12y = 0y(0) = 3y′(0) = 1
y(1) =
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + y′ + 25.25y = 25.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 25.25y = 25.25y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + y′ + 25.25y = 50.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 25.25y = 50.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. N = 10272
  2. 2. r = −2
  3. 3. k = 0.05062 min⁻¹
  4. 4. y(1) = 0.4641
  5. 5. ymax = 1.73
  6. 6. ymax = 3.461

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