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″ + 3y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3y′ + 25y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 8y′ + 15y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 15y = 0
r =
3.Medium
t (min)T (°C)37Ta​T = ?

A forged steel part at 340 °C is left to cool in a workshop at 21 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 37 min.

T0 = 340 °CTa = 21 °Ck = 0.05 min⁻¹t = 37 min
T = °C
4.Medium
t (min)T (°C)48Ta​80 °C

A forged steel part at 380 °C is left to cool in a workshop at 28 °C. After 48 min its temperature is 80 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 380 °CTa = 28 °CT(48) = 80 °C
k = min⁻¹
5.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 2y′ + 17y = 170 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 17y = 170y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 2y(0.5) = ?

A system obeys y″ + 2y′ + 17y = 34 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 2y′ + 17y = 34y(0) = 0y′(0) = 0
y(0.5) =

Answer key

  1. 1. ζ = 0.3
  2. 2. r = −3
  3. 3. T = 71.16 °C
  4. 4. k = 0.03984 min⁻¹
  5. 5. ymax = 14.56
  6. 6. y(0.5) = 2.229

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