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)N9 hN = ?

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.14 h⁻¹. How many cells N are there after 9 h?

N0 = 2000k = 0.14 h⁻¹t = 9 h
N =
2.Easy
ty1

A mass–spring–damper is described by y″ + 4y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4y′ + 100y = 0
ζ =
3.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = -2. Find y(1).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = -2
y(1) =
4.Medium
t (min)T (°C)12Ta​35 °C

A cup of coffee at 80 °C is left in a room at 18 °C. After 12 min its temperature is 35 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 80 °CTa = 18 °CT(12) = 35 °C
k = min⁻¹
5.Hard
tyy(0) = 1y(0.5) = ?

Solve y″ + y′ + 9.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + y′ + 9.25y = 0y(0) = 1y′(0) = 0
y(0.5) =
6.Hard
t (min)m (kg)t = ?4 kg

A tank holds 200 L of pure water. Brine with 0.05 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 4 kg of salt?

V = 200 Lm0 = 0 kgc = 0.05 kg/Lr = 5 L/min
t = min

Answer key

  1. 1. N = 7051
  2. 2. ζ = 0.2
  3. 3. y(1) = 0.8962
  4. 4. k = 0.1078 min⁻¹
  5. 5. y(0.5) = 0.1846
  6. 6. t = 20.43 min

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