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
ty1

A mass–spring–damper is described by y″ + 2y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2y′ + 100y = 0
ζ =
2.Easy
t (h)N5 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.28 h⁻¹. How many cells N are there after 5 h?

N0 = 500k = 0.28 h⁻¹t = 5 h
N =
3.Medium
t (min)T (°C)16Ta​T = ?

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 = 0.07 min⁻¹, find its temperature T after 16 min.

T0 = 85 °CTa = 20 °Ck = 0.07 min⁻¹t = 16 min
T = °C
4.Medium
t (min)T (°C)50Ta​T = ?

A forged steel part at 420 °C is left to cool in a workshop at 29 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 50 min.

T0 = 420 °CTa = 29 °Ck = 0.03 min⁻¹t = 50 min
T = °C
5.Hard
ty1y(1) = ?

Solve y′ + y = 5e−0.5t with y(0) = 4. Find y(1).

y′ + y = 5e−0.5ty(0) = 4
y(1) =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + y′ + 9.25y = 18.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 9.25y = 18.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.1
  2. 2. N = 2028
  3. 3. T = 41.21 °C
  4. 4. T = 116.2 °C
  5. 5. y(1) = 3.858
  6. 6. ymax = 3.185

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