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)N17 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 17 h?

N0 = 1500k = 0.36 h⁻¹t = 17 h
N =
2.Easy
rr² + br + c?

For the equation y″ + 8y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 12y = 0
r =
3.Medium
t (min)T (°C)55Ta​86 °C278 °C

A forged steel part at 360 °C is left to cool in a workshop at 21 °C. After 55 min its temperature is 86 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 278 °C?

T0 = 360 °CTa = 21 °CT(55) = 86 °CT = 278 °C
t = min
4.Medium
t (min)T (°C)14Ta​T = ?

A cup of coffee at 86 °C is left in a room at 24 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.11 min⁻¹, find its temperature T after 14 min.

T0 = 86 °CTa = 24 °Ck = 0.11 min⁻¹t = 14 min
T = °C
5.Hard
xy13(1, −3)y(3) = ?

Solve the initial value problem y′ + 3y/x = 3x2 for x > 0, with y(1) = −3. Find y(3).

y′ + 3y/x = 3x2y(1) = −3
y(3) =
6.Hard
tyy(0) = 2y(1) = ?

Solve y″ + 2y′ + 10y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 2y′ + 10y = 0y(0) = 2y′(0) = 0
y(1) =

Answer key

  1. 1. N = 682297
  2. 2. r = −2
  3. 3. t = 9.222 min
  4. 4. T = 37.29 °C
  5. 5. y(3) = 13.37
  6. 6. y(1) = −0.6938

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