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)22 dm = ?

A radioactive tracer has a half-life of 8 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 22 days?

m0 = 120 mgt½ = 8 dt = 22 d
m = mg
2.Easy
t (days)m (mg)22 dm = ?

A radioactive tracer has a half-life of 9 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 22 days?

m0 = 200 mgt½ = 9 dt = 22 d
m = mg
3.Medium
t (min)T (°C)15Ta​21 °C23 °C

A can of soda at 6 °C is taken out of the fridge into a room at 27 °C. After 15 min its temperature is 21 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 23 °C?

T0 = 6 °CTa = 27 °CT(15) = 21 °CT = 23 °C
t = min
4.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 1 and y′(0) = 2. Find y(1.5).

y″ + 7y′ + 10y = 0y(0) = 1y′(0) = 2
y(1.5) =
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 4y′ + 20y = 20 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 20y = 20y(0) = 0y′(0) = 0
ymax =
6.Hard
t (min)m (kg)t = ?26 kg

A tank holds 400 L of water in which 5 kg of salt is dissolved. Brine with 0.1 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 26 kg of salt?

V = 400 Lm0 = 5 kgc = 0.1 kg/Lr = 10 L/min
t = min

Answer key

  1. 1. m = 17.84 mg
  2. 2. m = 36.74 mg
  3. 3. t = 19.85 min
  4. 4. y(1.5) = 0.1154
  5. 5. ymax = 1.208
  6. 6. t = 36.65 min

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