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)y6 h560

A quantity grows according to dy/dt = k·y with k > 0. It is 200 at t = 0 and 560 at t = 6 h. Find the rate constant k in h⁻¹.

y(0) = 200y(6) = 560
k = h⁻¹
2.Easy
t (h)y5 h1000

A quantity grows according to dy/dt = k·y with k > 0. It is 250 at t = 0 and 1000 at t = 5 h. Find the rate constant k in h⁻¹.

y(0) = 250y(5) = 1000
k = h⁻¹
3.Medium
t (min)T (°C)20Ta​T = ?

A can of soda at 5 °C is taken out of the fridge into a room at 26 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 20 min.

T0 = 5 °CTa = 26 °Ck = 0.03 min⁻¹t = 20 min
T = °C
4.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 5 and y′(0) = 0. Find y(1).

y″ + 2y′ + y = 0y(0) = 5y′(0) = 0
y(1) =
5.Hard
t (min)m (kg)20 minm = ?

A tank holds 1000 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. How much salt m is in the tank after 20 min?

V = 1000 Lm0 = 0 kgc = 0.2 kg/Lr = 20 L/min
m = kg
6.Hard
t (min)m (kg)65 minm = ?

A tank holds 500 L of water in which 5 kg of salt is dissolved. Brine with 0.05 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 65 min?

V = 500 Lm0 = 5 kgc = 0.05 kg/Lr = 10 L/min
m = kg

Answer key

  1. 1. k = 0.1716 h⁻¹
  2. 2. k = 0.2773 h⁻¹
  3. 3. T = 14.47 °C
  4. 4. y(1) = 3.679
  5. 5. m = 65.94 kg
  6. 6. m = 19.55 kg

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