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″ + 9.6y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 9.6y′ + 64y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 3y′ + 2y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

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

Solve y″ + 8y′ + 16y = 0 with y(0) = 1 and y′(0) = 1. Find y(1.5).

y″ + 8y′ + 16y = 0y(0) = 1y′(0) = 1
y(1.5) =
4.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 2 and y′(0) = 4. Find y(0.5).

y″ + 6y′ + 8y = 0y(0) = 2y′(0) = 4
y(0.5) =
5.Hard
tyK = 5y(1) = ?

A system obeys y″ + 4y′ + 20y = 100 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 4y′ + 20y = 100y(0) = 0y′(0) = 0
y(1) =
6.Hard
tyy(0) = 3y(1.5) = ?

Solve y″ + 4y′ + 8y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 3y′(0) = 0
y(1.5) =

Answer key

  1. 1. ζ = 0.6
  2. 2. r = −1
  3. 3. y(1.5) = 0.02107
  4. 4. y(0.5) = 1.666
  5. 5. y(1) = 5.698
  6. 6. y(1.5) = −0.1268

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