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″ + 5y′ + 6y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 6y = 0
r =
2.Easy
ty1

A mass–spring–damper is described by y″ + 1.6y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.6y′ + 16y = 0
ζ =
3.Medium
tyy(0) = 2y(1) = ?

Solve y″ + 4y′ + 3y = 0 with y(0) = 2 and y′(0) = 1. Find y(1).

y″ + 4y′ + 3y = 0y(0) = 2y′(0) = 1
y(1) =
4.Medium
t (min)T (°C)30Ta​26 °C

A cup of coffee at 92 °C is left in a room at 24 °C. After 30 min its temperature is 26 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 92 °CTa = 24 °CT(30) = 26 °C
k = min⁻¹
5.Hard
tyy(0) = 2y(0.5) = ?

Solve y″ + 4y′ + 8y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 2y′(0) = 0
y(0.5) =
6.Hard
tyK = 10y(0.5) = ?

A system obeys y″ + y′ + 4.25y = 42.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + y′ + 4.25y = 42.5y(0) = 0y′(0) = 0
y(0.5) =

Answer key

  1. 1. r = −2
  2. 2. ζ = 0.2
  3. 3. y(1) = 1.213
  4. 4. k = 0.1175 min⁻¹
  5. 5. y(0.5) = 1.017
  6. 6. y(0.5) = 4.154

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