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.

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1.Easy
t (h)Nt = ?2000

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.33 h⁻¹. After how many hours t are there 2000 cells?

N0 = 200k = 0.33 h⁻¹N = 2000
t = h
2.Easy
t (h)y4 h200

A quantity grows according to dy/dt = k·y with k > 0. It is 100 at t = 0 and 200 at t = 4 h. Find the rate constant k in h⁻¹.

y(0) = 100y(4) = 200
k = h⁻¹
3.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 3 and y′(0) = 1. Find y(2).

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = 1
y(2) =
4.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 1 and y′(0) = -2. Find y(0.5).

y″ + 6y′ + 8y = 0y(0) = 1y′(0) = -2
y(0.5) =
5.Hard
ty2y(2) = ?

Solve y′ + y = 6e−2t with y(0) = 3. Find y(2).

y′ + y = 6e−2ty(0) = 3
y(2) =
6.Hard
tyK = 10ymax​ = ?

A system obeys y″ + y′ + 25.25y = 252.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 25.25y = 252.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. t = 6.978 h
  2. 2. k = 0.1733 h⁻¹
  3. 3. y(2) = 0.3114
  4. 4. y(0.5) = 0.3679
  5. 5. y(2) = 1.108
  6. 6. ymax = 17.3

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