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
ty1

A mass–spring–damper is described by y″ + 9.6y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 9.6y′ + 36y = 0
ζ =
2.Easy
t (h)y11 h204

A quantity grows according to dy/dt = k·y with k > 0. It is 120 at t = 0 and 204 at t = 11 h. Find the rate constant k in h⁻¹.

y(0) = 120y(11) = 204
k = h⁻¹
3.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 3 and y′(0) = 0. Find y(2).

y″ + 6y′ + 9y = 0y(0) = 3y′(0) = 0
y(2) =
4.Medium
tyy(0) = 2y(1) = ?

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

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

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

y′ + 2y/x = 4y(1) = −1
y(2) =
6.Hard
tyK = 1y(0.25) = ?

A system obeys y″ + y′ + 9.25y = 9.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + y′ + 9.25y = 9.25y(0) = 0y′(0) = 0
y(0.25) =

Answer key

  1. 1. ζ = 0.8
  2. 2. k = 0.04824 h⁻¹
  3. 3. y(2) = 0.05205
  4. 4. y(1) = 1.104
  5. 5. y(2) = 2.083
  6. 6. y(0.25) = 0.254

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