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 2000 cells and grows according to dN/dt = k·N with k = 0.17 h⁻¹. How many cells N are there after 23 h?

N₀ = 2000k = 0.17 h⁻¹t = 23 h

N =

2.Easy

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.4 h⁻¹. How many cells N are there after 7 h?

N₀ = 5000k = 0.4 h⁻¹t = 7 h

N =

3.Medium

A cup of coffee at 90 °C is left in a room at 21 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 6 min.

T₀ = 90 °CT_a = 21 °Ck = 0.03 min⁻¹t = 6 min

T = °C

4.Medium

A can of soda at 5 °C is taken out of the fridge into a room at 26 °C. After 11 min its temperature is 17 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 13 °C?

T₀ = 5 °CT_a = 26 °CT(11) = 17 °CT = 13 °C

t = min

5.Hard

A system obeys y″ + y′ + 4.25y = 42.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value y_max of its step response.

y″ + y′ + 4.25y = 42.5y(0) = 0y′(0) = 0

y_max =

6.Hard

Solve the initial value problem y′ + y/x = 3 for x > 0, with y(1) = 3. Find y(2).

y′ + y/x = 3y(1) = 3

y(2) =

Answer key

1. N = 99798
2. N = 82223
3. T = 78.63 °C
4. t = 6.226 min
5. y_max = 14.56
6. y(2) = 3.75

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