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″ + 6.4y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6.4y′ + 16y = 0
ζ =
3.Medium
t (min)T (°C)32Ta​254 °C151 °C

A forged steel part at 460 °C is left to cool in a workshop at 25 °C. After 32 min its temperature is 254 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 151 °C?

T0 = 460 °CTa = 25 °CT(32) = 254 °CT = 151 °C
t = min
4.Medium
tyy(0) = 4y(2) = ?

Solve y″ + 6y′ + 5y = 0 with y(0) = 4 and y′(0) = 1. Find y(2).

y″ + 6y′ + 5y = 0y(0) = 4y′(0) = 1
y(2) =
5.Hard
tyK = 1y(1) = ?

A system obeys y″ + y′ + 4.25y = 4.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + y′ + 4.25y = 4.25y(0) = 0y′(0) = 0
y(1) =
6.Hard
tyK = 5y(0.25) = ?

A system obeys y″ + 2y′ + 10y = 50 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + 2y′ + 10y = 50y(0) = 0y′(0) = 0
y(0.25) =

Answer key

  1. 1. r = −2
  2. 2. ζ = 0.8
  3. 3. t = 61.8 min
  4. 4. y(2) = 0.7105
  5. 5. y(1) = 1.115
  6. 6. y(0.25) = 1.266

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