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)N12 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.05 h⁻¹. How many cells N are there after 12 h?

N0 = 500k = 0.05 h⁻¹t = 12 h
N =
2.Easy
ty1

A mass–spring–damper is described by y″ + 4.2y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.2y′ + 9y = 0
ζ =
3.Medium
tyy(0) = 2y(2) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 2 and y′(0) = 2. Find y(2).

y″ + 7y′ + 12y = 0y(0) = 2y′(0) = 2
y(2) =
4.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 4y′ + 3y = 0 with y(0) = 5 and y′(0) = -2. Find y(1).

y″ + 4y′ + 3y = 0y(0) = 5y′(0) = -2
y(1) =
5.Hard
t (min)m (kg)10 minm = ?

A tank holds 1000 L of water in which 5 kg of salt is dissolved. Brine with 0.1 kg of salt per litre flows in at 20 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 10 min?

V = 1000 Lm0 = 5 kgc = 0.1 kg/Lr = 20 L/min
m = kg
6.Hard
t (min)m (kg)20 minm = ?

A tank holds 100 L of pure water. Brine with 0.25 kg of salt per litre flows in at 4 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 20 min?

V = 100 Lm0 = 0 kgc = 0.25 kg/Lr = 4 L/min
m = kg

Answer key

  1. 1. N = 911.1
  2. 2. ζ = 0.7
  3. 3. y(2) = 0.0221
  4. 4. y(1) = 2.317
  5. 5. m = 22.22 kg
  6. 6. m = 13.77 kg

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