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″ + 7.2y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 7.2y′ + 36y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 14.4y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 14.4y′ + 36y = 0
ζ =
3.Medium
t (min)T (°C)27Ta​31 °C25 °C

A cup of coffee at 85 °C is left in a room at 18 °C. After 27 min its temperature is 31 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 25 °C?

T0 = 85 °CTa = 18 °CT(27) = 31 °CT = 25 °C
t = min
4.Medium
tyy(0) = 1y(1) = ?

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

y″ + 5y′ + 4y = 0y(0) = 1y′(0) = -2
y(1) =
5.Hard
tyK = 5y(1) = ?

A system obeys y″ + 4y′ + 8y = 40 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 4y′ + 8y = 40y(0) = 0y′(0) = 0
y(1) =
6.Hard
t (min)m (kg)20 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 10 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 = 5 kgc = 0.1 kg/Lr = 10 L/min
m = kg

Answer key

  1. 1. ζ = 0.6
  2. 2. ζ = 1.2
  3. 3. t = 37.19 min
  4. 4. y(1) = 0.2514
  5. 5. y(1) = 4.666
  6. 6. m = 9.323 kg

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