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 = ?400

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.04 h⁻¹. After how many hours t are there 400 cells?

N0 = 200k = 0.04 h⁻¹N = 400
t = h
2.Easy
t (days)m (mg)15 dm = ?

A radioactive tracer has a half-life of 4 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 15 days?

m0 = 200 mgt½ = 4 dt = 15 d
m = mg
3.Medium
t (min)T (°C)26Ta​14 °C

A can of soda at 5 °C is taken out of the fridge into a room at 22 °C. After 26 min its temperature is 14 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 5 °CTa = 22 °CT(26) = 14 °C
k = min⁻¹
4.Medium
t (min)T (°C)14Ta​36 °C

A cup of coffee at 86 °C is left in a room at 20 °C. After 14 min its temperature is 36 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 86 °CTa = 20 °CT(14) = 36 °C
k = min⁻¹
5.Hard
xy13(1, −2)y(3) = ?

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

y′ + y/x = 1y(1) = −2
y(3) =
6.Hard
tyy(0) = 5y(0.75) = ?

Solve y″ + 3y′ + 6.25y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 3y′ + 6.25y = 0y(0) = 5y′(0) = 0
y(0.75) =

Answer key

  1. 1. t = 17.33 h
  2. 2. m = 14.87 mg
  3. 3. k = 0.02899 min⁻¹
  4. 4. k = 0.1012 min⁻¹
  5. 5. y(3) = 0.6667
  6. 6. y(0.75) = 1.329

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