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
ty1

A mass–spring–damper is described by y″ + 2.5y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.5y′ + 25y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 8y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 8y′ + 25y = 0
ζ =
3.Medium
tyy(0) = 1y(1) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 1 and y′(0) = -2. Find y(1).

y″ + 6y′ + 9y = 0y(0) = 1y′(0) = -2
y(1) =
4.Medium
tyy(0) = 1y(2) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 1 and y′(0) = 6. Find y(2).

y″ + 5y′ + 6y = 0y(0) = 1y′(0) = 6
y(2) =
5.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 2y′ + 5y = 25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 5y = 25y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 1y(0.5) = ?

A system obeys y″ + 3y′ + 11.25y = 11.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 3y′ + 11.25y = 11.25y(0) = 0y′(0) = 0
y(0.5) =

Answer key

  1. 1. ζ = 0.25
  2. 2. ζ = 0.8
  3. 3. y(1) = 0.09957
  4. 4. y(2) = 0.145
  5. 5. ymax = 6.039
  6. 6. y(0.5) = 0.731

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