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 decays according to dy/dt = −k·y with k > 0. It is 80 at t = 0 and 16 at t = 6 h. Find the rate constant k in h⁻¹.

y(0) = 80y(6) = 16

k = h⁻¹

2.Easy

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.16 h⁻¹. How many cells N are there after 19 h?

N₀ = 2000k = 0.16 h⁻¹t = 19 h

N =

3.Medium

A can of soda at 3 °C is taken out of the fridge into a room at 27 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 5 min.

T₀ = 3 °CT_a = 27 °Ck = 0.04 min⁻¹t = 5 min

T = °C

4.Medium

Solve y″ + 7y′ + 10y = 0 with y(0) = 4 and y′(0) = 0. Find y(1.5).

y″ + 7y′ + 10y = 0y(0) = 4y′(0) = 0

y(1.5) =

5.Hard

A system obeys y″ + y′ + 4.25y = 4.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + y′ + 4.25y = 4.25y(0) = 0y′(0) = 0

y(0.75) =

6.Hard

A system obeys y″ + 3y′ + 27.25y = 136.3 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 3y′ + 27.25y = 136.3y(0) = 0y′(0) = 0

y(1) =

Answer key

1. k = 0.2682 h⁻¹
2. N = 41810
3. T = 7.35 °C
4. y(1.5) = 0.3304
5. y(0.75) = 0.78
6. y(1) = 5.004

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