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 = ?6000

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.24 h⁻¹. After how many hours t are there 6000 cells?

N0 = 2000k = 0.24 h⁻¹N = 6000
t = h
2.Easy
ty1

A mass–spring–damper is described by y″ + 5y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 5y′ + 100y = 0
ζ =
3.Medium
tyy(0) = 2y(2) = ?

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

y″ + 3y′ + 2y = 0y(0) = 2y′(0) = 4
y(2) =
4.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) =
5.Hard
xy12(1, 3)y(2) = ?

Solve the initial value problem y′ + y/x = 3 for x > 0, with y(1) = 3. Find y(2).

y′ + y/x = 3y(1) = 3
y(2) =
6.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 2y′ + 5y = 50 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 5y = 50y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. t = 4.578 h
  2. 2. ζ = 0.25
  3. 3. y(2) = 0.9728
  4. 4. y(1.5) = 3.012
  5. 5. y(2) = 3.75
  6. 6. ymax = 12.08

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