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″ + 0.8y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 0.8y′ + 16y = 0
ζ =
3.Medium
t (min)T (°C)22Ta​22 °C9 °C

A can of soda at 4 °C is taken out of the fridge into a room at 26 °C. After 22 min its temperature is 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 9 °C?

T0 = 4 °CTa = 26 °CT(22) = 22 °CT = 9 °C
t = min
4.Medium
t (min)T (°C)24Ta​50 °C73 °C

A cup of coffee at 95 °C is left in a room at 22 °C. After 24 min its temperature is 50 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 73 °C?

T0 = 95 °CTa = 22 °CT(24) = 50 °CT = 73 °C
t = min
5.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 4y′ + 29y = 58 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 29y = 58y(0) = 0y′(0) = 0
ymax =
6.Hard
tyy(0) = 2y(1) = ?

Solve y″ + 3y′ + 18.25y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 3y′ + 18.25y = 0y(0) = 2y′(0) = 0
y(1) =

Answer key

  1. 1. r = −2
  2. 2. ζ = 0.1
  3. 3. t = 3.327 min
  4. 4. t = 8.982 min
  5. 5. ymax = 2.569
  6. 6. y(1) = −0.4183

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