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″ + 2.4y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 36y = 0
ζ =
2.Easy
t (h)N3 hN = ?

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.13 h⁻¹. How many cells N are there after 3 h?

N0 = 1000k = 0.13 h⁻¹t = 3 h
N =
3.Medium
t (min)T (°C)26Ta​T = ?

A cup of coffee at 95 °C is left in a room at 24 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.06 min⁻¹, find its temperature T after 26 min.

T0 = 95 °CTa = 24 °Ck = 0.06 min⁻¹t = 26 min
T = °C
4.Medium
t (min)T (°C)16Ta​T = ?

A can of soda at 3 °C is taken out of the fridge into a room at 28 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 16 min.

T0 = 3 °CTa = 28 °Ck = 0.03 min⁻¹t = 16 min
T = °C
5.Hard
xy13(1, 3)y(3) = ?

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

y′ + 2y/x = 3y(1) = 3
y(3) =
6.Hard
ty2y(2) = ?

Solve y′ + 0.5y = 10e−2t with y(0) = 0. Find y(2).

y′ + 0.5y = 10e−2ty(0) = 0
y(2) =

Answer key

  1. 1. ζ = 0.2
  2. 2. N = 1477
  3. 3. T = 38.92 °C
  4. 4. T = 12.53 °C
  5. 5. y(3) = 3.222
  6. 6. y(2) = 2.33

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