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″ + 6.4y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6.4y′ + 64y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 4.8y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 36y = 0
ζ =
3.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 5 and y′(0) = 2. Find y(0.5).

y″ + 8y′ + 16y = 0y(0) = 5y′(0) = 2
y(0.5) =
4.Medium
t (min)T (°C)19Ta​T = ?

A forged steel part at 320 °C is left to cool in a workshop at 27 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.02 min⁻¹, find its temperature T after 19 min.

T0 = 320 °CTa = 27 °Ck = 0.02 min⁻¹t = 19 min
T = °C
5.Hard
tyy(0) = 1y(0.5) = ?

Solve y″ + 3y′ + 27.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 3y′ + 27.25y = 0y(0) = 1y′(0) = 0
y(0.5) =
6.Hard
ty1.5y(1.5) = ?

Solve y′ + 0.5y = 8e−3t with y(0) = 5. Find y(1.5).

y′ + 0.5y = 8e−3ty(0) = 5
y(1.5) =

Answer key

  1. 1. ζ = 0.4
  2. 2. ζ = 0.4
  3. 3. y(0.5) = 2.165
  4. 4. T = 227.4 °C
  5. 5. y(0.5) = −0.2936
  6. 6. y(1.5) = 3.838

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