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)Nt = ?10000

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.28 h⁻¹. After how many hours t are there 10000 cells?

N0 = 2000k = 0.28 h⁻¹N = 10000
t = h
2.Easy
ty1

A mass–spring–damper is described by y″ + 4.2y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.2y′ + 9y = 0
ζ =
3.Medium
t (min)T (°C)24Ta​T = ?

A can of soda at 5 °C is taken out of the fridge into 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 24 min.

T0 = 5 °CTa = 23 °Ck = 0.06 min⁻¹t = 24 min
T = °C
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 5 and y′(0) = 2. Find y(2).

y″ + 4y′ + 4y = 0y(0) = 5y′(0) = 2
y(2) =
5.Hard
t (min)m (kg)t = ?10 kg

A tank holds 100 L of water in which 20 kg of salt is dissolved. Brine with 0.05 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 10 kg of salt?

V = 100 Lm0 = 20 kgc = 0.05 kg/Lr = 10 L/min
t = min
6.Hard
tyK = 2y(1) = ?

A system obeys y″ + y′ + 16.25y = 32.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + y′ + 16.25y = 32.5y(0) = 0y′(0) = 0
y(1) =

Answer key

  1. 1. t = 5.748 h
  2. 2. ζ = 0.7
  3. 3. T = 18.74 °C
  4. 4. y(2) = 0.5312
  5. 5. t = 10.99 min
  6. 6. y(1) = 2.908

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