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″ + 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 =
2.Easy
ty1

A mass–spring–damper is described by y″ + 4.8y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 36y = 0
ζ =
3.Medium
tyy(0) = 2y(2) = ?

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

y″ + 4y′ + 4y = 0y(0) = 2y′(0) = -2
y(2) =
4.Medium
t (min)T (°C)12Ta​T = ?

A forged steel part at 380 °C is left to cool in a workshop at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 12 min.

T0 = 380 °CTa = 22 °Ck = 0.03 min⁻¹t = 12 min
T = °C
5.Hard
t (min)m (kg)t = ?100 kg

A tank holds 1000 L of pure water. Brine with 0.2 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 100 kg of salt?

V = 1000 Lm0 = 0 kgc = 0.2 kg/Lr = 10 L/min
t = min
6.Hard
t (min)m (kg)60 minm = ?

A tank holds 400 L of water in which 5 kg of salt is dissolved. Brine with 0.1 kg of salt per litre flows in at 4 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 60 min?

V = 400 Lm0 = 5 kgc = 0.1 kg/Lr = 4 L/min
m = kg

Answer key

  1. 1. r = −3
  2. 2. ζ = 0.4
  3. 3. y(2) = 0.1099
  4. 4. T = 271.8 °C
  5. 5. t = 69.31 min
  6. 6. m = 20.79 kg

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