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″ + 0.8y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 0.8y′ + 4y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

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

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

y″ + 6y′ + 9y = 0y(0) = 2y′(0) = 0
y(2) =
4.Medium
t (min)T (°C)12Ta​14 °C21 °C

A can of soda at 6 °C is taken out of the fridge into a room at 26 °C. After 12 min its temperature is 14 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 21 °C?

T0 = 6 °CTa = 26 °CT(12) = 14 °CT = 21 °C
t = min
5.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 3y′ + 6.25y = 12.5 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 = 12.5y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + y′ + 25.25y = 50.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 25.25y = 50.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.2
  2. 2. r = −3
  3. 3. y(2) = 0.0347
  4. 4. t = 32.57 min
  5. 5. ymax = 2.19
  6. 6. ymax = 3.461

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