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″ + 3.6y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3.6y′ + 9y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 6.4y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6.4y′ + 64y = 0
ζ =
3.Medium
tyy(0) = 2y(2) = ?

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

y″ + 2y′ + y = 0y(0) = 2y′(0) = -2
y(2) =
4.Medium
t (min)T (°C)28Ta​48 °C64 °C

A cup of coffee at 87 °C is left in a room at 18 °C. After 28 min its temperature is 48 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 64 °C?

T0 = 87 °CTa = 18 °CT(28) = 48 °CT = 64 °C
t = min
5.Hard
t (min)m (kg)65 minm = ?

A tank holds 1000 L of water in which 10 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 2 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 = 1000 Lm0 = 10 kgc = 0.25 kg/Lr = 2 L/min
m = kg
6.Hard
tyK = 10ymax​ = ?

A system obeys y″ + y′ + 16.25y = 162.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 16.25y = 162.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.6
  2. 2. ζ = 0.4
  3. 3. y(2) = 0.2707
  4. 4. t = 13.63 min
  5. 5. m = 39.26 kg
  6. 6. ymax = 16.75

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