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
rr² + br + c?

For the equation y″ + 7y′ + 10y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 10y = 0
r =
2.Easy
ty1

A mass–spring–damper is described by y″ + 9.6y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 9.6y′ + 16y = 0
ζ =
3.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 3 and y′(0) = 2. Find y(0.5).

y″ + 8y′ + 16y = 0y(0) = 3y′(0) = 2
y(0.5) =
4.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 3y′ + 2y = 0 with y(0) = 3 and y′(0) = 6. Find y(2).

y″ + 3y′ + 2y = 0y(0) = 3y′(0) = 6
y(2) =
5.Hard
ty0.5y(0.5) = ?

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

y′ + 3y = 3e−ty(0) = 2
y(0.5) =
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + y′ + 25.25y = 25.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 25.25y = 25.25y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. r = −2
  2. 2. ζ = 1.2
  3. 3. y(0.5) = 1.353
  4. 4. y(2) = 1.459
  5. 5. y(0.5) = 1.021
  6. 6. ymax = 1.73

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