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″ + 8y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 8y′ + 25y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 8.4y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 8.4y′ + 36y = 0
ζ =
3.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 5 and y′(0) = 6. Find y(2).

y″ + 8y′ + 15y = 0y(0) = 5y′(0) = 6
y(2) =
4.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 1 and y′(0) = 4. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 1y′(0) = 4
y(1.5) =
5.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 =
6.Hard
ty0.5y(0.5) = ?

Solve y′ + 0.5y = 10e−2t with y(0) = 3. Find y(0.5).

y′ + 0.5y = 10e−2ty(0) = 3
y(0.5) =

Answer key

  1. 1. ζ = 0.8
  2. 2. ζ = 0.7
  3. 3. y(2) = 0.03794
  4. 4. y(1.5) = 1.897
  5. 5. ymax = 12.08
  6. 6. y(0.5) = 5.076

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