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 = ?7500

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.15 h⁻¹. After how many hours t are there 7500 cells?

N0 = 1500k = 0.15 h⁻¹N = 7500
t = h
2.Easy
t (h)N15 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.13 h⁻¹. How many cells N are there after 15 h?

N0 = 500k = 0.13 h⁻¹t = 15 h
N =
3.Medium
tyy(0) = 2y(1) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 2 and y′(0) = 0. Find y(1).

y″ + 6y′ + 9y = 0y(0) = 2y′(0) = 0
y(1) =
4.Medium
t (min)T (°C)26Ta​25 °C

A can of soda at 4 °C is taken out of the fridge into a room at 27 °C. After 26 min its temperature is 25 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 4 °CTa = 27 °CT(26) = 25 °C
k = min⁻¹
5.Hard
tyK = 1y(1.5) = ?

A system obeys y″ + 4y′ + 29y = 29 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 4y′ + 29y = 29y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
xy13(1, −3)y(3) = ?

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

y′ + 3y/x = 1y(1) = −3
y(3) =

Answer key

  1. 1. t = 10.73 h
  2. 2. N = 3514
  3. 3. y(1) = 0.3983
  4. 4. k = 0.09394 min⁻¹
  5. 5. y(1.5) = 0.9641
  6. 6. y(3) = 0.6296

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