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″ + 1.5y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.5y′ + 9y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 12y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 12y′ + 100y = 0
ζ =
3.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 3 and y′(0) = -1. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 3y′(0) = -1
y(1.5) =
4.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 3 and y′(0) = -2. Find y(0.5).

y″ + 8y′ + 15y = 0y(0) = 3y′(0) = -2
y(0.5) =
5.Hard
tyK = 2y(0.25) = ?

A system obeys y″ + y′ + 9.25y = 18.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + y′ + 9.25y = 18.5y(0) = 0y′(0) = 0
y(0.25) =
6.Hard
tyy(0) = 1y(1.5) = ?

Solve y″ + 3y′ + 6.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 3y′ + 6.25y = 0y(0) = 1y′(0) = 0
y(1.5) =

Answer key

  1. 1. ζ = 0.25
  2. 2. ζ = 0.6
  3. 3. y(1.5) = 1.339
  4. 4. y(0.5) = 1.163
  5. 5. y(0.25) = 0.5081
  6. 6. y(1.5) = −0.09319

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