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″ + 2.4y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 9y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 4y′ + 3y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 4y′ + 3y = 0
r =
3.Medium
t (min)T (°C)18Ta​405 °C

A forged steel part at 480 °C is left to cool in a workshop at 26 °C. After 18 min its temperature is 405 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 480 °CTa = 26 °CT(18) = 405 °C
k = min⁻¹
4.Medium
t (min)T (°C)15Ta​65 °C36 °C

A cup of coffee at 91 °C is left in a room at 20 °C. After 15 min its temperature is 65 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 36 °C?

T0 = 91 °CTa = 20 °CT(15) = 65 °CT = 36 °C
t = min
5.Hard
t (min)m (kg)t = ?16 kg

A tank holds 200 L of water in which 5 kg of salt is dissolved. Brine with 0.1 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 16 kg of salt?

V = 200 Lm0 = 5 kgc = 0.1 kg/Lr = 2 L/min
t = min
6.Hard
ty0.5y(0.5) = ?

Solve y′ + 3y = 2e−0.5t with y(0) = 1. Find y(0.5).

y′ + 3y = 2e−0.5ty(0) = 1
y(0.5) =

Answer key

  1. 1. ζ = 0.4
  2. 2. r = −1
  3. 3. k = 0.01003 min⁻¹
  4. 4. t = 49.01 min
  5. 5. t = 132.2 min
  6. 6. y(0.5) = 0.6677

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