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 16 days. A sample initially contains 40 mg. The decay follows dm/dt = −k·m. How much m remains after 29 days?

m₀ = 40 mgt_½ = 16 dt = 29 d

m = mg

2.Easy

A mass–spring–damper is described by y″ + 6y′ + 4y = 0. Compare it with y″ + 2ζω_ny′ + ω_n²y = 0 and find the damping ratio ζ.

y″ + 6y′ + 4y = 0

ζ =

3.Medium

A can of soda at 5 °C is taken out of the fridge into a room at 30 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 26 min.

T₀ = 5 °CT_a = 30 °Ck = 0.03 min⁻¹t = 26 min

T = °C

4.Medium

A can of soda at 6 °C is taken out of the fridge into a room at 22 °C. After 17 min its temperature is 12 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 8 °C?

T₀ = 6 °CT_a = 22 °CT(17) = 12 °CT = 8 °C

t = min

5.Hard

A tank holds 400 L of water in which 10 kg of salt is dissolved. Brine with 0.1 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 25 min?

V = 400 Lm₀ = 10 kgc = 0.1 kg/Lr = 20 L/min

m = kg

6.Hard

Solve y″ + 3y′ + 27.25y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 3y′ + 27.25y = 0y(0) = 10y′(0) = 0

y(0.25) =

Answer key

1. m = 11.39 mg
2. ζ = 1.5
3. T = 18.54 °C
4. t = 4.83 min
5. m = 31.4 kg
6. y(0.25) = 4.124

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