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
rr² + br + c?

For the equation y″ + 6y′ + 8y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 8y = 0
r =
3.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 3 and y′(0) = -4. Find y(2).

y″ + 7y′ + 12y = 0y(0) = 3y′(0) = -4
y(2) =
4.Medium
tyy(0) = 1y(2) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 1 and y′(0) = 0. Find y(2).

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

Solve y″ + 2y′ + 5y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 2y′ + 5y = 0y(0) = 5y′(0) = 0
y(1.5) =
6.Hard
xy12(1, 1)y(2) = ?

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

y′ + y/x = 3y(1) = 1
y(2) =

Answer key

  1. 1. r = −2
  2. 2. r = −2
  3. 3. y(2) = 0.01815
  4. 4. y(2) = 0.04999
  5. 5. y(1.5) = −1.026
  6. 6. y(2) = 2.75

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