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)Nt = ?50000

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.13 h⁻¹. After how many hours t are there 50000 cells?

N0 = 5000k = 0.13 h⁻¹N = 50000
t = h
2.Easy
ty1

A mass–spring–damper is described by y″ + 4.8y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 16y = 0
ζ =
3.Medium
t (min)T (°C)13Ta​246 °C

A forged steel part at 400 °C is left to cool in a workshop at 20 °C. After 13 min its temperature is 246 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 400 °CTa = 20 °CT(13) = 246 °C
k = min⁻¹
4.Medium
tyy(0) = 1y(2) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 1 and y′(0) = 2. Find y(2).

y″ + 4y′ + 4y = 0y(0) = 1y′(0) = 2
y(2) =
5.Hard
tyK = 2y(0.75) = ?

A system obeys y″ + 2y′ + 17y = 34 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 2y′ + 17y = 34y(0) = 0y′(0) = 0
y(0.75) =
6.Hard
xy12(1, −3)y(2) = ?

Solve the initial value problem y′ + 2y/x = 4x2 for x > 0, with y(1) = −3. Find y(2).

y′ + 2y/x = 4x2y(1) = −3
y(2) =

Answer key

  1. 1. t = 17.71 h
  2. 2. ζ = 0.6
  3. 3. k = 0.03997 min⁻¹
  4. 4. y(2) = 0.1648
  5. 5. y(0.75) = 2.902
  6. 6. y(2) = 5.45

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