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″ + 15y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 15y′ + 25y = 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 = 0.36 h⁻¹. After how many hours t are there 3000 cells?

N0 = 1500k = 0.36 h⁻¹N = 3000
t = h
3.Medium
t (min)T (°C)12Ta​44 °C

A cup of coffee at 89 °C is left in a room at 24 °C. After 12 min its temperature is 44 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 89 °CTa = 24 °CT(12) = 44 °C
k = min⁻¹
4.Medium
t (min)T (°C)5Ta​63 °C

A cup of coffee at 86 °C is left in a room at 23 °C. After 5 min its temperature is 63 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 86 °CTa = 23 °CT(5) = 63 °C
k = min⁻¹
5.Hard
xy12(1, 3)y(2) = ?

Solve the initial value problem y′ + 2y/x = 5 for x > 0, with y(1) = 3. Find y(2).

y′ + 2y/x = 5y(1) = 3
y(2) =
6.Hard
ty1y(1) = ?

Solve y′ + y = 9e−3t with y(0) = 4. Find y(1).

y′ + y = 9e−3ty(0) = 4
y(1) =

Answer key

  1. 1. ζ = 1.5
  2. 2. t = 1.925 h
  3. 3. k = 0.09822 min⁻¹
  4. 4. k = 0.09085 min⁻¹
  5. 5. y(2) = 3.667
  6. 6. y(1) = 2.903

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