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)77 dm = ?

A radioactive tracer has a half-life of 21 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 77 days?

m0 = 200 mgt½ = 21 dt = 77 d
m = mg
2.Easy
t (h)Nt = ?5000

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.06 h⁻¹. After how many hours t are there 5000 cells?

N0 = 1000k = 0.06 h⁻¹N = 5000
t = h
3.Medium
t (min)T (°C)6Ta​T = ?

A can of soda at 5 °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 = 0.09 min⁻¹, find its temperature T after 6 min.

T0 = 5 °CTa = 28 °Ck = 0.09 min⁻¹t = 6 min
T = °C
4.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 2 and y′(0) = 2. Find y(0.5).

y″ + 6y′ + 9y = 0y(0) = 2y′(0) = 2
y(0.5) =
5.Hard
xy13(1, −2)y(3) = ?

Solve the initial value problem y′ + 3y/x = 3x2 for x > 0, with y(1) = −2. Find y(3).

y′ + 3y/x = 3x2y(1) = −2
y(3) =
6.Hard
t (min)m (kg)t = ?10 kg

A tank holds 200 L of pure water. Brine with 0.1 kg of salt per litre flows in at 4 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 10 kg of salt?

V = 200 Lm0 = 0 kgc = 0.1 kg/Lr = 4 L/min
t = min

Answer key

  1. 1. m = 15.75 mg
  2. 2. t = 26.82 h
  3. 3. T = 14.6 °C
  4. 4. y(0.5) = 1.339
  5. 5. y(3) = 13.41
  6. 6. t = 34.66 min

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