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)26 dm = ?

A radioactive tracer has a half-life of 20 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 26 days?

m0 = 10 mgt½ = 20 dt = 26 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 12y = 0
r =
3.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 3 and y′(0) = -1. Find y(0.5).

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = -1
y(0.5) =
4.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 2 and y′(0) = 0. Find y(0.5).

y″ + 8y′ + 16y = 0y(0) = 2y′(0) = 0
y(0.5) =
5.Hard
t (min)m (kg)t = ?56 kg

A tank holds 400 L of pure water. Brine with 0.2 kg of salt per litre flows in at 20 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 56 kg of salt?

V = 400 Lm0 = 0 kgc = 0.2 kg/Lr = 20 L/min
t = min
6.Hard
t (min)m (kg)t = ?75 kg

A tank holds 1000 L of pure water. Brine with 0.25 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 75 kg of salt?

V = 1000 Lm0 = 0 kgc = 0.25 kg/Lr = 5 L/min
t = min

Answer key

  1. 1. m = 4.061 mg
  2. 2. r = −3
  3. 3. y(0.5) = 2.023
  4. 4. y(0.5) = 0.812
  5. 5. t = 24.08 min
  6. 6. t = 71.33 min

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