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″ + y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + y′ + 25y = 0
ζ =
2.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 =
3.Medium
t (min)T (°C)29Ta​24 °C73 °C

A cup of coffee at 94 °C is left in a room at 20 °C. After 29 min its temperature is 24 °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 = 94 °CTa = 20 °CT(29) = 24 °CT = 73 °C
t = min
4.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 1 and y′(0) = 1. Find y(1.5).

y″ + 8y′ + 16y = 0y(0) = 1y′(0) = 1
y(1.5) =
5.Hard
xy13(1, 3)y(3) = ?

Solve the initial value problem y′ + y/x = 3 for x > 0, with y(1) = 3. Find y(3).

y′ + y/x = 3y(1) = 3
y(3) =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 4y′ + 8y = 16 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 8y = 16y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.1
  2. 2. r = −2
  3. 3. t = 3.317 min
  4. 4. y(1.5) = 0.02107
  5. 5. y(3) = 5
  6. 6. ymax = 2.086

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