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″ + 10y′ + 24y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 10y′ + 24y = 0
r =
2.Easy
rr² + br + c?

For the equation y″ + 4y′ + 3y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 4y′ + 3y = 0
r =
3.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 4y′ + 3y = 0 with y(0) = 1 and y′(0) = 0. Find y(1.5).

y″ + 4y′ + 3y = 0y(0) = 1y′(0) = 0
y(1.5) =
4.Medium
tyy(0) = 4y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 4 and y′(0) = -4. Find y(0.5).

y″ + 8y′ + 15y = 0y(0) = 4y′(0) = -4
y(0.5) =
5.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 3y′ + 27.25y = 272.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 27.25y = 272.5y(0) = 0y′(0) = 0
ymax =
6.Hard
tyy(0) = 3y(0.5) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 3y′(0) = 0
y(0.5) =

Answer key

  1. 1. r = −4
  2. 2. r = −1
  3. 3. y(1.5) = 0.3291
  4. 4. y(0.5) = 1.457
  5. 5. ymax = 13.9
  6. 6. y(0.5) = −1.24

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