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″ + 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 =
2.Easy
t (h)Nt = ?15000

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.06 h⁻¹. After how many hours t are there 15000 cells?

N0 = 1500k = 0.06 h⁻¹N = 15000
t = h
3.Medium
tyy(0) = 2y(1) = ?

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

y″ + 4y′ + 4y = 0y(0) = 2y′(0) = 4
y(1) =
4.Medium
t (min)T (°C)50Ta​88 °C58 °C

A forged steel part at 480 °C is left to cool in a workshop at 27 °C. After 50 min its temperature is 88 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 58 °C?

T0 = 480 °CTa = 27 °CT(50) = 88 °CT = 58 °C
t = min
5.Hard
t (min)m (kg)10 minm = ?

A tank holds 1000 L of pure water. Brine with 0.25 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 10 min?

V = 1000 Lm0 = 0 kgc = 0.25 kg/Lr = 2 L/min
m = kg
6.Hard
tyK = 5y(1) = ?

A system obeys y″ + 4y′ + 29y = 145 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 4y′ + 29y = 145y(0) = 0y′(0) = 0
y(1) =

Answer key

  1. 1. r = −2
  2. 2. t = 38.38 h
  3. 3. y(1) = 1.353
  4. 4. t = 66.88 min
  5. 5. m = 4.95 kg
  6. 6. y(1) = 5.068

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