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)N22 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.17 h⁻¹. How many cells N are there after 22 h?

N0 = 500k = 0.17 h⁻¹t = 22 h
N =
2.Easy
ty1

A mass–spring–damper is described by y″ + 3.2y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3.2y′ + 4y = 0
ζ =
3.Medium
t (min)T (°C)16Ta​35 °C

A cup of coffee at 82 °C is left in a room at 20 °C. After 16 min its temperature is 35 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 82 °CTa = 20 °CT(16) = 35 °C
k = min⁻¹
4.Medium
t (min)T (°C)8Ta​T = ?

A can of soda at 4 °C is taken out of the fridge into a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.07 min⁻¹, find its temperature T after 8 min.

T0 = 4 °CTa = 22 °Ck = 0.07 min⁻¹t = 8 min
T = °C
5.Hard
tyy(0) = 5y(0.75) = ?

Solve y″ + 3y′ + 6.25y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 3y′ + 6.25y = 0y(0) = 5y′(0) = 0
y(0.75) =
6.Hard
t (min)m (kg)65 minm = ?

A tank holds 200 L of water in which 10 kg of salt is dissolved. Brine with 0.1 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 65 min?

V = 200 Lm0 = 10 kgc = 0.1 kg/Lr = 5 L/min
m = kg

Answer key

  1. 1. N = 21049
  2. 2. ζ = 0.8
  3. 3. k = 0.08869 min⁻¹
  4. 4. T = 11.72 °C
  5. 5. y(0.75) = 1.329
  6. 6. m = 18.03 kg

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