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

y″ + 2y′ + 25y = 0
ζ =
3.Medium
tyy(0) = 4y(1) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 4 and y′(0) = 1. Find y(1).

y″ + 7y′ + 12y = 0y(0) = 4y′(0) = 1
y(1) =
4.Medium
t (min)T (°C)20Ta​42 °C

A cup of coffee at 86 °C is left in a room at 23 °C. After 20 min its temperature is 42 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 86 °CTa = 23 °CT(20) = 42 °C
k = min⁻¹
5.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 2y′ + 10y = 100 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 10y = 100y(0) = 0y′(0) = 0
ymax =
6.Hard
t (min)m (kg)55 minm = ?

A tank holds 200 L of pure water. Brine with 0.05 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 55 min?

V = 200 Lm0 = 0 kgc = 0.05 kg/Lr = 5 L/min
m = kg

Answer key

  1. 1. r = −2
  2. 2. ζ = 0.2
  3. 3. y(1) = 0.6083
  4. 4. k = 0.05993 min⁻¹
  5. 5. ymax = 13.51
  6. 6. m = 7.472 kg

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