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″ + 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 =
2.Easy
ty1

A mass–spring–damper is described by y″ + 6.4y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6.4y′ + 16y = 0
ζ =
3.Medium
tyy(0) = 3y(1.5) = ?

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

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

Solve y″ + 2y′ + y = 0 with y(0) = 2 and y′(0) = -1. Find y(0.5).

y″ + 2y′ + y = 0y(0) = 2y′(0) = -1
y(0.5) =
5.Hard
xy12(1, −2)y(2) = ?

Solve the initial value problem y′ + 2y/x = 4x2 for x > 0, with y(1) = −2. Find y(2).

y′ + 2y/x = 4x2y(1) = −2
y(2) =
6.Hard
xy13(1, 2)y(3) = ?

Solve the initial value problem y′ + y/x = 6x for x > 0, with y(1) = 2. Find y(3).

y′ + y/x = 6xy(1) = 2
y(3) =

Answer key

  1. 1. r = −1
  2. 2. ζ = 0.8
  3. 3. y(1.5) = 1.673
  4. 4. y(0.5) = 1.516
  5. 5. y(2) = 5.7
  6. 6. y(3) = 18

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