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″ + 4.8y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 4y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 9y′ + 20y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 9y′ + 20y = 0
r =
3.Medium
t (min)T (°C)20Ta​T = ?

A cup of coffee at 95 °C is left in a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.08 min⁻¹, find its temperature T after 20 min.

T0 = 95 °CTa = 22 °Ck = 0.08 min⁻¹t = 20 min
T = °C
4.Medium
tyy(0) = 1y(1) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 1 and y′(0) = 2. Find y(1).

y″ + 6y′ + 8y = 0y(0) = 1y′(0) = 2
y(1) =
5.Hard
t (min)m (kg)25 minm = ?

A tank holds 500 L of water in which 5 kg of salt is dissolved. Brine with 0.2 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 25 min?

V = 500 Lm0 = 5 kgc = 0.2 kg/Lr = 2 L/min
m = kg
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + y′ + 9.25y = 18.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 9.25y = 18.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 1.2
  2. 2. r = −4
  3. 3. T = 36.74 °C
  4. 4. y(1) = 0.3694
  5. 5. m = 14.04 kg
  6. 6. ymax = 3.185

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