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″ + 4y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4y′ + 16y = 0
ζ =
2.Easy
t (h)Nt = ?3000

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.15 h⁻¹. After how many hours t are there 3000 cells?

N0 = 1500k = 0.15 h⁻¹N = 3000
t = h
3.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 1 and y′(0) = 2. Find y(1.5).

y″ + 7y′ + 12y = 0y(0) = 1y′(0) = 2
y(1.5) =
4.Medium
t (min)T (°C)10Ta​62 °C71 °C

A cup of coffee at 91 °C is left in a room at 18 °C. After 10 min its temperature is 62 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 71 °C?

T0 = 91 °CTa = 18 °CT(10) = 62 °CT = 71 °C
t = min
5.Hard
xy12(1, 5)y(2) = ?

Solve the initial value problem y′ + 2y/x = 2x for x > 0, with y(1) = 5. Find y(2).

y′ + 2y/x = 2xy(1) = 5
y(2) =
6.Hard
tyy(0) = 3y(0.25) = ?

Solve y″ + 4y′ + 8y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 3y′(0) = 0
y(0.25) =

Answer key

  1. 1. ζ = 0.5
  2. 2. t = 4.621 h
  3. 3. y(1.5) = 0.05426
  4. 4. t = 6.324 min
  5. 5. y(2) = 3.125
  6. 6. y(0.25) = 2.469

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