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″ + 3.2y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3.2y′ + 4y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 3y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3y′ + 9y = 0
ζ =
3.Medium
t (min)T (°C)25Ta​T = ?

A cup of coffee at 88 °C is left in a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.11 min⁻¹, find its temperature T after 25 min.

T0 = 88 °CTa = 22 °Ck = 0.11 min⁻¹t = 25 min
T = °C
4.Medium
tyy(0) = 4y(1) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 4 and y′(0) = 0. Find y(1).

y″ + 7y′ + 12y = 0y(0) = 4y′(0) = 0
y(1) =
5.Hard
ty1y(1) = ?

Solve y′ + 2y = 9e−t with y(0) = 5. Find y(1).

y′ + 2y = 9e−ty(0) = 5
y(1) =
6.Hard
tyK = 1y(1) = ?

A system obeys y″ + 4y′ + 29y = 29 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 4y′ + 29y = 29y(0) = 0y′(0) = 0
y(1) =

Answer key

  1. 1. ζ = 0.8
  2. 2. ζ = 0.5
  3. 3. T = 26.22 °C
  4. 4. y(1) = 0.5768
  5. 5. y(1) = 2.77
  6. 6. y(1) = 1.014

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