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)N23 hN = ?

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.38 h⁻¹. How many cells N are there after 23 h?

N0 = 200k = 0.38 h⁻¹t = 23 h
N =
2.Easy
ty1

A mass–spring–damper is described by y″ + 2.4y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 9y = 0
ζ =
3.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 1 and y′(0) = -2. Find y(1.5).

y″ + 8y′ + 15y = 0y(0) = 1y′(0) = -2
y(1.5) =
4.Medium
tyy(0) = 4y(1) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 4 and y′(0) = 0. Find y(1).

y″ + 7y′ + 10y = 0y(0) = 4y′(0) = 0
y(1) =
5.Hard
t (min)m (kg)50 minm = ?

A tank holds 100 L of water in which 5 kg of salt is dissolved. Brine with 0.1 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 50 min?

V = 100 Lm0 = 5 kgc = 0.1 kg/Lr = 4 L/min
m = kg
6.Hard
tyK = 10y(0.5) = ?

A system obeys y″ + 3y′ + 18.25y = 182.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 3y′ + 18.25y = 182.5y(0) = 0y′(0) = 0
y(0.5) =

Answer key

  1. 1. N = 1.25 × 10⁶
  2. 2. ζ = 0.4
  3. 3. y(1.5) = 0.01639
  4. 4. y(1) = 0.8843
  5. 5. m = 9.323 kg
  6. 6. y(0.5) = 10.36

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