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)y2 h90

A quantity decays according to dy/dt = −k·y with k > 0. It is 120 at t = 0 and 90 at t = 2 h. Find the rate constant k in h⁻¹.

y(0) = 120y(2) = 90
k = h⁻¹
2.Easy
t (h)y7 h1400

A quantity grows according to dy/dt = k·y with k > 0. It is 400 at t = 0 and 1400 at t = 7 h. Find the rate constant k in h⁻¹.

y(0) = 400y(7) = 1400
k = h⁻¹
3.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 1 and y′(0) = -1. Find y(0.5).

y″ + 6y′ + 9y = 0y(0) = 1y′(0) = -1
y(0.5) =
4.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 6y′ + 5y = 0 with y(0) = 2 and y′(0) = 6. Find y(1.5).

y″ + 6y′ + 5y = 0y(0) = 2y′(0) = 6
y(1.5) =
5.Hard
t (min)m (kg)t = ?30 kg

A tank holds 500 L of water in which 10 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 30 kg of salt?

V = 500 Lm0 = 10 kgc = 0.1 kg/Lr = 10 L/min
t = min
6.Hard
t (min)m (kg)t = ?146 kg

A tank holds 1000 L of water in which 20 kg of salt is dissolved. Brine with 0.2 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 146 kg of salt?

V = 1000 Lm0 = 20 kgc = 0.2 kg/Lr = 2 L/min
t = min

Answer key

  1. 1. k = 0.1438 h⁻¹
  2. 2. k = 0.179 h⁻¹
  3. 3. y(0.5) = 0.4463
  4. 4. y(1.5) = 0.8914
  5. 5. t = 34.66 min
  6. 6. t = 602 min

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