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 15 days. A sample initially contains 50 mg. The decay follows dm/dt = −k·m. How much m remains after 24 days?

m₀ = 50 mgt_½ = 15 dt = 24 d

m = mg

2.Easy

A quantity decays according to dy/dt = −k·y with k > 0. It is 120 at t = 0 and 24 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 120y(12) = 24

k = h⁻¹

3.Medium

Solve y″ + 7y′ + 12y = 0 with y(0) = 1 and y′(0) = -2. Find y(1.5).

y″ + 7y′ + 12y = 0y(0) = 1y′(0) = -2

y(1.5) =

4.Medium

A can of soda at 4 °C is taken out of the fridge into a room at 25 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 22 min.

T₀ = 4 °CT_a = 25 °Ck = 0.04 min⁻¹t = 22 min

T = °C

5.Hard

Solve y′ + 2y = 4e^−t with y(0) = 3. Find y(0.5).

y′ + 2y = 4e^−ty(0) = 3

y(0.5) =

6.Hard

A system obeys y″ + 4y′ + 20y = 20 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 4y′ + 20y = 20y(0) = 0y′(0) = 0

y(0.75) =

Answer key

1. m = 16.49 mg
2. k = 0.1341 h⁻¹
3. y(1.5) = 0.01974
4. T = 16.29 °C
5. y(0.5) = 2.058
6. y(0.75) = 1.205

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