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″ + 2.4y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 4y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 2.4y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 16y = 0
ζ =
3.Medium
t (min)T (°C)23Ta​25 °C

A can of soda at 6 °C is taken out of the fridge into a room at 28 °C. After 23 min its temperature is 25 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 6 °CTa = 28 °CT(23) = 25 °C
k = min⁻¹
4.Medium
tyy(0) = 2y(2) = ?

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

y″ + 2y′ + y = 0y(0) = 2y′(0) = 1
y(2) =
5.Hard
ty1.5y(1.5) = ?

Solve y′ + 3y = 10e−2t with y(0) = 3. Find y(1.5).

y′ + 3y = 10e−2ty(0) = 3
y(1.5) =
6.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 =

Answer key

  1. 1. ζ = 0.6
  2. 2. ζ = 0.3
  3. 3. k = 0.08663 min⁻¹
  4. 4. y(2) = 1.083
  5. 5. y(1.5) = 0.4201
  6. 6. ymax = 13.9

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