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 = ?10000

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.34 h⁻¹. After how many hours t are there 10000 cells?

N0 = 2000k = 0.34 h⁻¹N = 10000
t = h
2.Easy
t (h)Nt = ?1000

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.17 h⁻¹. After how many hours t are there 1000 cells?

N0 = 200k = 0.17 h⁻¹N = 1000
t = h
3.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 6y′ + 5y = 0 with y(0) = 5 and y′(0) = -2. Find y(1).

y″ + 6y′ + 5y = 0y(0) = 5y′(0) = -2
y(1) =
4.Medium
t (min)T (°C)24Ta​20 °C

A can of soda at 3 °C is taken out of the fridge into a room at 30 °C. After 24 min its temperature is 20 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 3 °CTa = 30 °CT(24) = 20 °C
k = min⁻¹
5.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 2y′ + 10y = 100 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 10y = 100y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 5y(1.5) = ?

A system obeys y″ + 4y′ + 13y = 65 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 4y′ + 13y = 65y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. t = 4.734 h
  2. 2. t = 9.467 h
  3. 3. y(1) = 2.11
  4. 4. k = 0.04139 min⁻¹
  5. 5. ymax = 13.51
  6. 6. y(1.5) = 5.215

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