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)N16 hN = ?

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.36 h⁻¹. How many cells N are there after 16 h?

N0 = 1500k = 0.36 h⁻¹t = 16 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)7Ta​15 °C

A can of soda at 5 °C is taken out of the fridge into a room at 25 °C. After 7 min its temperature is 15 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 5 °CTa = 25 °CT(7) = 15 °C
k = min⁻¹
4.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 1 and y′(0) = 2. Find y(0.5).

y″ + 8y′ + 15y = 0y(0) = 1y′(0) = 2
y(0.5) =
5.Hard
xy13(1, 5)y(3) = ?

Solve the initial value problem y′ + 2y/x = x for x > 0, with y(1) = 5. Find y(3).

y′ + 2y/x = xy(1) = 5
y(3) =
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 2y′ + 5y = 5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 5y = 5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. N = 476022
  2. 2. r = −2
  3. 3. k = 0.09902 min⁻¹
  4. 4. y(0.5) = 0.5757
  5. 5. y(3) = 2.778
  6. 6. ymax = 1.208

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