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″ + 6y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6y′ + 100y = 0
ζ =
2.Easy
t (h)N11 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.21 h⁻¹. How many cells N are there after 11 h?

N0 = 5000k = 0.21 h⁻¹t = 11 h
N =
3.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 3 and y′(0) = 4. Find y(1.5).

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = 4
y(1.5) =
4.Medium
tyy(0) = 3y(1) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 3 and y′(0) = 1. Find y(1).

y″ + 5y′ + 6y = 0y(0) = 3y′(0) = 1
y(1) =
5.Hard
t (min)m (kg)t = ?12 kg

A tank holds 200 L of pure water. Brine with 0.2 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 12 kg of salt?

V = 200 Lm0 = 0 kgc = 0.2 kg/Lr = 5 L/min
t = min
6.Hard
tyK = 10y(1.5) = ?

A system obeys y″ + 2y′ + 17y = 170 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 2y′ + 17y = 170y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. ζ = 0.3
  2. 2. N = 50372
  3. 3. y(1.5) = 0.8962
  4. 4. y(1) = 1.005
  5. 5. t = 14.27 min
  6. 6. y(1.5) = 8.013

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