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
rr² + br + c?

For the equation y″ + 6y′ + 8y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 8y = 0
r =
2.Easy
ty1

A mass–spring–damper is described by y″ + 2.4y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 4y = 0
ζ =
3.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 2 and y′(0) = 2. Find y(0.5).

y″ + 7y′ + 10y = 0y(0) = 2y′(0) = 2
y(0.5) =
4.Medium
t (min)T (°C)28Ta​184 °C

A forged steel part at 300 °C is left to cool in a workshop at 29 °C. After 28 min its temperature is 184 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 300 °CTa = 29 °CT(28) = 184 °C
k = min⁻¹
5.Hard
t (min)m (kg)t = ?12 kg

A tank holds 400 L of pure water. Brine with 0.05 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 12 kg of salt?

V = 400 Lm0 = 0 kgc = 0.05 kg/Lr = 20 L/min
t = min
6.Hard
tyy(0) = 5y(0.75) = ?

Solve y″ + 2y′ + 5y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 2y′ + 5y = 0y(0) = 5y′(0) = 0
y(0.75) =

Answer key

  1. 1. r = −2
  2. 2. ζ = 0.6
  3. 3. y(0.5) = 1.307
  4. 4. k = 0.01995 min⁻¹
  5. 5. t = 18.33 min
  6. 6. y(0.75) = 1.345

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