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″ + 2y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2y′ + 25y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 4.8y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 36y = 0
ζ =
3.Medium
tyy(0) = 5y(1.5) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 5 and y′(0) = 0. Find y(1.5).

y″ + 5y′ + 6y = 0y(0) = 5y′(0) = 0
y(1.5) =
4.Medium
tyy(0) = 5y(1) = ?

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

y″ + 2y′ + y = 0y(0) = 5y′(0) = 0
y(1) =
5.Hard
tyK = 2y(0.75) = ?

A system obeys y″ + y′ + 4.25y = 8.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + y′ + 4.25y = 8.5y(0) = 0y′(0) = 0
y(0.75) =
6.Hard
tyy(0) = 10y(0.5) = ?

Solve y″ + 2y′ + 5y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 2y′ + 5y = 0y(0) = 10y′(0) = 0
y(0.5) =

Answer key

  1. 1. ζ = 0.2
  2. 2. ζ = 0.4
  3. 3. y(1.5) = 0.6357
  4. 4. y(1) = 3.679
  5. 5. y(0.75) = 1.56
  6. 6. y(0.5) = 5.829

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