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
t (h)N20 hN = ?

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 20 h?

N0 = 200k = 0.26 h⁻¹t = 20 h
N =
2.Easy
ty1

A mass–spring–damper is described by y″ + 4.8y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 9y = 0
ζ =
3.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 2 and y′(0) = 2. Find y(0.5).

y″ + 8y′ + 16y = 0y(0) = 2y′(0) = 2
y(0.5) =
4.Medium
t (min)T (°C)5Ta​10 °C

A can of soda at 6 °C is taken out of the fridge into a room at 24 °C. After 5 min its temperature is 10 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 6 °CTa = 24 °CT(5) = 10 °C
k = min⁻¹
5.Hard
tyK = 2ymax​ = ?

A system obeys y″ + y′ + 25.25y = 50.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 25.25y = 50.5y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 10y(1.5) = ?

A system obeys y″ + 2y′ + 26y = 260 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 2y′ + 26y = 260y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. N = 36254
  2. 2. ζ = 0.8
  3. 3. y(0.5) = 0.9473
  4. 4. k = 0.05026 min⁻¹
  5. 5. ymax = 3.461
  6. 6. y(1.5) = 8.808

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