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″ + 3.2y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3.2y′ + 16y = 0
ζ =
3.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = 0. Find y(0.5).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = 0
y(0.5) =
4.Medium
t (min)T (°C)49Ta​161 °C89 °C

A forged steel part at 380 °C is left to cool in a workshop at 30 °C. After 49 min its temperature is 161 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 89 °C?

T0 = 380 °CTa = 30 °CT(49) = 161 °CT = 89 °C
t = min
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 3y′ + 11.25y = 11.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 11.25y = 11.25y(0) = 0y′(0) = 0
ymax =
6.Hard
tyy(0) = 2y(1) = ?

Solve y″ + 4y′ + 8y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 2y′(0) = 0
y(1) =

Answer key

  1. 1. r = −3
  2. 2. ζ = 0.4
  3. 3. y(0.5) = 2.789
  4. 4. t = 88.77 min
  5. 5. ymax = 1.208
  6. 6. y(1) = 0.1335

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