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″ + 3y′ + 2y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 3y′ + 2y = 0
r =
2.Easy
ty1

A mass–spring–damper is described by y″ + 1.2y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.2y′ + 36y = 0
ζ =
3.Medium
tyy(0) = 1y(2) = ?

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

y″ + 4y′ + 4y = 0y(0) = 1y′(0) = -1
y(2) =
4.Medium
t (min)T (°C)38Ta​T = ?

A forged steel part at 440 °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 = 0.04 min⁻¹, find its temperature T after 38 min.

T0 = 440 °CTa = 22 °Ck = 0.04 min⁻¹t = 38 min
T = °C
5.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 4y′ + 29y = 145 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 29y = 145y(0) = 0y′(0) = 0
ymax =
6.Hard
t (min)m (kg)t = ?37 kg

A tank holds 400 L of water in which 10 kg of salt is dissolved. Brine with 0.25 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 37 kg of salt?

V = 400 Lm0 = 10 kgc = 0.25 kg/Lr = 10 L/min
t = min

Answer key

  1. 1. r = −1
  2. 2. ζ = 0.1
  3. 3. y(2) = 0.05495
  4. 4. T = 113.4 °C
  5. 5. ymax = 6.423
  6. 6. t = 14.27 min

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