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)N24 hN = ?

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.02 h⁻¹. How many cells N are there after 24 h?

N0 = 200k = 0.02 h⁻¹t = 24 h
N =
2.Easy
ty1

A mass–spring–damper is described by y″ + 19.2y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 19.2y′ + 64y = 0
ζ =
3.Medium
tyy(0) = 3y(1) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 3 and y′(0) = 6. Find y(1).

y″ + 6y′ + 8y = 0y(0) = 3y′(0) = 6
y(1) =
4.Medium
tyy(0) = 3y(1) = ?

Solve y″ + 5y′ + 4y = 0 with y(0) = 3 and y′(0) = 6. Find y(1).

y″ + 5y′ + 4y = 0y(0) = 3y′(0) = 6
y(1) =
5.Hard
tyK = 10ymax​ = ?

A system obeys y″ + y′ + 4.25y = 42.5 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 = 42.5y(0) = 0y′(0) = 0
ymax =
6.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 =

Answer key

  1. 1. N = 323.2
  2. 2. ζ = 1.2
  3. 3. y(1) = 1.108
  4. 4. y(1) = 2.152
  5. 5. ymax = 14.56
  6. 6. ymax = 7.28

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