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″ + 12y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 12y′ + 100y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 14y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 14y′ + 100y = 0
ζ =
3.Medium
t (min)T (°C)58Ta​T = ?

A forged steel part at 380 °C is left to cool in a workshop at 26 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.01 min⁻¹, find its temperature T after 58 min.

T0 = 380 °CTa = 26 °Ck = 0.01 min⁻¹t = 58 min
T = °C
4.Medium
t (min)T (°C)7Ta​13 °C23 °C

A can of soda at 3 °C is taken out of the fridge into a room at 27 °C. After 7 min its temperature is 13 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 23 °C?

T0 = 3 °CTa = 27 °CT(7) = 13 °CT = 23 °C
t = min
5.Hard
xy13(1, 0)y(3) = ?

Solve the initial value problem y′ + 3y/x = 6 for x > 0, with y(1) = 0. Find y(3).

y′ + 3y/x = 6y(1) = 0
y(3) =
6.Hard
tyy(0) = 10y(0.75) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 10y′(0) = 0
y(0.75) =

Answer key

  1. 1. ζ = 0.6
  2. 2. ζ = 0.7
  3. 3. T = 224.2 °C
  4. 4. t = 23.27 min
  5. 5. y(3) = 4.444
  6. 6. y(0.75) = −4.416

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