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
t (h)y6 h80

A quantity decays according to dy/dt = −k·y with k > 0. It is 400 at t = 0 and 80 at t = 6 h. Find the rate constant k in h⁻¹.

y(0) = 400y(6) = 80
k = h⁻¹
2.Easy
rr² + br + c?

For the equation y″ + 9y′ + 18y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 9y′ + 18y = 0
r =
3.Medium
tyy(0) = 4y(1.5) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 4 and y′(0) = 0. Find y(1.5).

y″ + 7y′ + 12y = 0y(0) = 4y′(0) = 0
y(1.5) =
4.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 2 and y′(0) = -2. Find y(1.5).

y″ + 8y′ + 16y = 0y(0) = 2y′(0) = -2
y(1.5) =
5.Hard
tyy(0) = 3y(1.5) = ?

Solve y″ + y′ + 25.25y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + y′ + 25.25y = 0y(0) = 3y′(0) = 0
y(1.5) =
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 3y′ + 6.25y = 6.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 6.25y = 6.25y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. k = 0.2682 h⁻¹
  2. 2. r = −3
  3. 3. y(1.5) = 0.148
  4. 4. y(1.5) = 0.02727
  5. 5. y(1.5) = 0.6241
  6. 6. ymax = 1.095

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