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

A quantity grows according to dy/dt = k·y with k > 0. It is 400 at t = 0 and 600 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 400y(12) = 600

k = h⁻¹

2.Easy

For the equation y″ + 9y′ + 18y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 9y′ + 18y = 0

r =

3.Medium

A cup of coffee at 91 °C is left in a room at 22 °C. After 5 min its temperature is 60 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T₀ = 91 °CT_a = 22 °CT(5) = 60 °C

k = min⁻¹

4.Medium

A can of soda at 6 °C is taken out of the fridge into a room at 22 °C. After 19 min its temperature is 20 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T₀ = 6 °CT_a = 22 °CT(19) = 20 °C

k = min⁻¹

5.Hard

A tank holds 1000 L of water in which 20 kg of salt is dissolved. Brine with 0.25 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 15 min?

V = 1000 Lm₀ = 20 kgc = 0.25 kg/Lr = 10 L/min

m = kg

6.Hard

A system obeys y″ + 4y′ + 13y = 65 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 4y′ + 13y = 65y(0) = 0y′(0) = 0

y(0.5) =

Answer key

1. k = 0.03379 h⁻¹
2. r = −3
3. k = 0.1193 min⁻¹
4. k = 0.1094 min⁻¹
5. m = 52.04 kg
6. y(0.5) = 3.647

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