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 = ?2500

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.12 h⁻¹. After how many hours t are there 2500 cells?

N0 = 500k = 0.12 h⁻¹N = 2500
t = h
2.Easy
t (h)N9 hN = ?

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.22 h⁻¹. How many cells N are there after 9 h?

N0 = 200k = 0.22 h⁻¹t = 9 h
N =
3.Medium
t (min)T (°C)11Ta​T = ?

A forged steel part at 440 °C is left to cool in a workshop at 25 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 11 min.

T0 = 440 °CTa = 25 °Ck = 0.04 min⁻¹t = 11 min
T = °C
4.Medium
t (min)T (°C)27Ta​T = ?

A can of soda at 3 °C is taken out of the fridge into a room at 30 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.06 min⁻¹, find its temperature T after 27 min.

T0 = 3 °CTa = 30 °Ck = 0.06 min⁻¹t = 27 min
T = °C
5.Hard
tyK = 5y(0.75) = ?

A system obeys y″ + 2y′ + 5y = 25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 2y′ + 5y = 25y(0) = 0y′(0) = 0
y(0.75) =
6.Hard
xy12(1, −2)y(2) = ?

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

y′ + y/x = 3xy(1) = −2
y(2) =

Answer key

  1. 1. t = 13.41 h
  2. 2. N = 1449
  3. 3. T = 292.3 °C
  4. 4. T = 24.66 °C
  5. 5. y(0.75) = 3.655
  6. 6. y(2) = 2.5

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