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
t (h)N3 hN = ?

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.38 h⁻¹. How many cells N are there after 3 h?

N0 = 2000k = 0.38 h⁻¹t = 3 h
N =
2.Easy
ty1

A mass–spring–damper is described by y″ + 12y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 12y′ + 25y = 0
ζ =
3.Medium
t (min)T (°C)24Ta​17 °C

A can of soda at 5 °C is taken out of the fridge into a room at 28 °C. After 24 min its temperature is 17 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 5 °CTa = 28 °CT(24) = 17 °C
k = min⁻¹
4.Medium
t (min)T (°C)36Ta​156 °C102 °C

A forged steel part at 420 °C is left to cool in a workshop at 20 °C. After 36 min its temperature is 156 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 102 °C?

T0 = 420 °CTa = 20 °CT(36) = 156 °CT = 102 °C
t = min
5.Hard
tyK = 5ymax​ = ?

A system obeys y″ + y′ + 25.25y = 126.3 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 = 126.3y(0) = 0y′(0) = 0
ymax =
6.Hard
xy13(1, 3)y(3) = ?

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

y′ + y/x = 3y(1) = 3
y(3) =

Answer key

  1. 1. N = 6254
  2. 2. ζ = 1.2
  3. 3. k = 0.03073 min⁻¹
  4. 4. t = 52.88 min
  5. 5. ymax = 8.652
  6. 6. y(3) = 5

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