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 (days)m (mg)14 dm = ?

A radioactive tracer has a half-life of 13 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 14 days?

m0 = 20 mgt½ = 13 dt = 14 d
m = mg
2.Easy
t (h)Nt = ?10000

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.29 h⁻¹. After how many hours t are there 10000 cells?

N0 = 2000k = 0.29 h⁻¹N = 10000
t = h
3.Medium
t (min)T (°C)21Ta​29 °C64 °C

A cup of coffee at 87 °C is left in a room at 23 °C. After 21 min its temperature is 29 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 64 °C?

T0 = 87 °CTa = 23 °CT(21) = 29 °CT = 64 °C
t = min
4.Medium
t (min)T (°C)8Ta​T = ?

A can of soda at 4 °C is taken out of the fridge into a room at 28 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 8 min.

T0 = 4 °CTa = 28 °Ck = 0.03 min⁻¹t = 8 min
T = °C
5.Hard
tyK = 5y(0.5) = ?

A system obeys y″ + 4y′ + 20y = 100 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 4y′ + 20y = 100y(0) = 0y′(0) = 0
y(0.5) =
6.Hard
ty1y(1) = ?

Solve y′ + 3y = 5e−2t with y(0) = 2. Find y(1).

y′ + 3y = 5e−2ty(0) = 2
y(1) =

Answer key

  1. 1. m = 9.481 mg
  2. 2. t = 5.55 h
  3. 3. t = 3.951 min
  4. 4. T = 9.121 °C
  5. 5. y(0.5) = 4.929
  6. 6. y(1) = 0.5273

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