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 bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.26 h⁻¹. How many cells N are there after 5 h?

N₀ = 200k = 0.26 h⁻¹t = 5 h

N =

2.Easy

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.15 h⁻¹. After how many hours t are there 2000 cells?

N₀ = 1000k = 0.15 h⁻¹N = 2000

t = h

3.Medium

A cup of coffee at 85 °C is left in a room at 20 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 29 min.

T₀ = 85 °CT_a = 20 °Ck = 0.04 min⁻¹t = 29 min

T = °C

4.Medium

Solve y″ + 7y′ + 10y = 0 with y(0) = 4 and y′(0) = 1. Find y(0.5).

y″ + 7y′ + 10y = 0y(0) = 4y′(0) = 1

y(0.5) =

5.Hard

Solve y″ + 2y′ + 5y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 2y′ + 5y = 0y(0) = 5y′(0) = 0

y(1.5) =

6.Hard

A system obeys y″ + 3y′ + 18.25y = 18.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value y_max of its step response.

y″ + 3y′ + 18.25y = 18.25y(0) = 0y′(0) = 0

y_max =

Answer key

1. N = 733.9
2. t = 4.621 h
3. T = 40.38 °C
4. y(0.5) = 2.329
5. y(1.5) = −1.026
6. y_max = 1.308

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