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
rr² + br + c?

For the equation y″ + 5y′ + 4y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 4y = 0
r =
2.Easy
t (h)N24 hN = ?

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.24 h⁻¹. How many cells N are there after 24 h?

N0 = 2000k = 0.24 h⁻¹t = 24 h
N =
3.Medium
tyy(0) = 2y(1) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 2 and y′(0) = 4. Find y(1).

y″ + 8y′ + 15y = 0y(0) = 2y′(0) = 4
y(1) =
4.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 5 and y′(0) = -1. Find y(0.5).

y″ + 8y′ + 16y = 0y(0) = 5y′(0) = -1
y(0.5) =
5.Hard
t (min)m (kg)t = ?50 kg

A tank holds 400 L of water in which 5 kg of salt is dissolved. Brine with 0.2 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 50 kg of salt?

V = 400 Lm0 = 5 kgc = 0.2 kg/Lr = 10 L/min
t = min
6.Hard
t (min)m (kg)115 minm = ?

A tank holds 400 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. How much salt m is in the tank after 115 min?

V = 400 Lm0 = 0 kgc = 0.05 kg/Lr = 5 L/min
m = kg

Answer key

  1. 1. r = −1
  2. 2. N = 634697
  3. 3. y(1) = 0.3148
  4. 4. y(0.5) = 1.962
  5. 5. t = 36.65 min
  6. 6. m = 15.25 kg

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