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)Nt = ?1000

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.22 h⁻¹. After how many hours t are there 1000 cells?

N0 = 200k = 0.22 h⁻¹N = 1000
t = h
2.Easy
ty1

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

y″ + 2y′ + 100y = 0
ζ =
3.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = -2. Find y(0.5).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = -2
y(0.5) =
4.Medium
t (min)T (°C)17Ta​T = ?

A cup of coffee at 95 °C is left in a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.12 min⁻¹, find its temperature T after 17 min.

T0 = 95 °CTa = 22 °Ck = 0.12 min⁻¹t = 17 min
T = °C
5.Hard
xy13(1, 1)y(3) = ?

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

y′ + y/x = 1y(1) = 1
y(3) =
6.Hard
xy12(1, 0)y(2) = ?

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

y′ + y/x = 3xy(1) = 0
y(2) =

Answer key

  1. 1. t = 7.316 h
  2. 2. ζ = 0.1
  3. 3. y(0.5) = 2.566
  4. 4. T = 31.49 °C
  5. 5. y(3) = 1.667
  6. 6. y(2) = 3.5

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