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 radioactive tracer has a half-life of 4 days. A sample initially contains 100 mg. The decay follows dm/dt = −k·m. How much m remains after 9 days?

m₀ = 100 mgt_½ = 4 dt = 9 d

m = mg

2.Easy

A quantity decays according to dy/dt = −k·y with k > 0. It is 200 at t = 0 and 140 at t = 8 h. Find the rate constant k in h⁻¹.

y(0) = 200y(8) = 140

k = h⁻¹

3.Medium

A can of soda at 5 °C is taken out of the fridge into a room at 25 °C. After 22 min its temperature is 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 14 °C?

T₀ = 5 °CT_a = 25 °CT(22) = 22 °CT = 14 °C

t = min

4.Medium

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = 2. Find y(1).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = 2

y(1) =

5.Hard

A tank holds 100 L of water in which 20 kg of salt is dissolved. Brine with 0.1 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 12 kg of salt?

V = 100 Lm₀ = 20 kgc = 0.1 kg/Lr = 5 L/min

t = min

6.Hard

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 70 min?

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

m = kg

Answer key

1. m = 21.02 mg
2. k = 0.04458 h⁻¹
3. t = 6.933 min
4. y(1) = 1.095
5. t = 32.19 min
6. m = 20.07 kg

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