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″ + 12y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 12y′ + 25y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 12y = 0
r =
3.Medium
t (min)T (°C)53Ta​82 °C

A forged steel part at 480 °C is left to cool in a workshop at 28 °C. After 53 min its temperature is 82 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 480 °CTa = 28 °CT(53) = 82 °C
k = min⁻¹
4.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 3 and y′(0) = 2. Find y(0.5).

y″ + 8y′ + 16y = 0y(0) = 3y′(0) = 2
y(0.5) =
5.Hard
t (min)m (kg)t = ?89 kg

A tank holds 1000 L of water in which 20 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 20 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 89 kg of salt?

V = 1000 Lm0 = 20 kgc = 0.25 kg/Lr = 20 L/min
t = min
6.Hard
xy12(1, −3)y(2) = ?

Solve the initial value problem y′ + 2y/x = 2x2 for x > 0, with y(1) = −3. Find y(2).

y′ + 2y/x = 2x2y(1) = −3
y(2) =

Answer key

  1. 1. ζ = 1.2
  2. 2. r = −3
  3. 3. k = 0.04009 min⁻¹
  4. 4. y(0.5) = 1.353
  5. 5. t = 17.83 min
  6. 6. y(2) = 2.35

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