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″ + 3y′ + 2y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 3y′ + 2y = 0
r =
2.Easy
t (h)Nt = ?50000

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.03 h⁻¹. After how many hours t are there 50000 cells?

N0 = 5000k = 0.03 h⁻¹N = 50000
t = h
3.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 3 and y′(0) = 4. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 3y′(0) = 4
y(1.5) =
4.Medium
tyy(0) = 5y(1.5) = ?

Solve y″ + 4y′ + 3y = 0 with y(0) = 5 and y′(0) = -2. Find y(1.5).

y″ + 4y′ + 3y = 0y(0) = 5y′(0) = -2
y(1.5) =
5.Hard
tyK = 1y(1) = ?

A system obeys y″ + y′ + 9.25y = 9.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + y′ + 9.25y = 9.25y(0) = 0y′(0) = 0
y(1) =
6.Hard
t (min)m (kg)t = ?40 kg

A tank holds 1000 L of pure water. Brine with 0.1 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 40 kg of salt?

V = 1000 Lm0 = 0 kgc = 0.1 kg/Lr = 10 L/min
t = min

Answer key

  1. 1. r = −1
  2. 2. t = 76.75 h
  3. 3. y(1.5) = 3.012
  4. 4. y(1.5) = 1.434
  5. 5. y(1) = 1.586
  6. 6. t = 51.08 min

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