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)N2 hN = ?

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.26 h⁻¹. How many cells N are there after 2 h?

N0 = 2000k = 0.26 h⁻¹t = 2 h
N =
2.Easy
t (days)m (mg)71 dm = ?

A radioactive tracer has a half-life of 25 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 71 days?

m0 = 120 mgt½ = 25 dt = 71 d
m = mg
3.Medium
tyy(0) = 5y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 5 and y′(0) = 1. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 5y′(0) = 1
y(1.5) =
4.Medium
t (min)T (°C)24Ta​38 °C64 °C

A cup of coffee at 89 °C is left in a room at 22 °C. After 24 min its temperature is 38 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 64 °C?

T0 = 89 °CTa = 22 °CT(24) = 38 °CT = 64 °C
t = min
5.Hard
tyK = 5ymax​ = ?

A system obeys y″ + y′ + 4.25y = 21.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 4.25y = 21.25y(0) = 0y′(0) = 0
ymax =
6.Hard
ty1.5y(1.5) = ?

Solve y′ + 3y = 4e−2t with y(0) = 1. Find y(1.5).

y′ + 3y = 4e−2ty(0) = 1
y(1.5) =

Answer key

  1. 1. N = 3364
  2. 2. m = 16.76 mg
  3. 3. y(1.5) = 3.124
  4. 4. t = 7.827 min
  5. 5. ymax = 7.28
  6. 6. y(1.5) = 0.1658

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