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″ + 8y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 8y′ + 64y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 12y = 0
r =
3.Medium
t (min)T (°C)8Ta​61 °C79 °C

A cup of coffee at 87 °C is left in a room at 20 °C. After 8 min its temperature is 61 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 79 °C?

T0 = 87 °CTa = 20 °CT(8) = 61 °CT = 79 °C
t = min
4.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 2 and y′(0) = 1. Find y(1.5).

y″ + 6y′ + 9y = 0y(0) = 2y′(0) = 1
y(1.5) =
5.Hard
xy12(1, −1)y(2) = ?

Solve the initial value problem y′ + 2y/x = 5 for x > 0, with y(1) = −1. Find y(2).

y′ + 2y/x = 5y(1) = −1
y(2) =
6.Hard
ty1y(1) = ?

Solve y′ + 3y = 9e−t with y(0) = 0. Find y(1).

y′ + 3y = 9e−ty(0) = 0
y(1) =

Answer key

  1. 1. ζ = 0.5
  2. 2. r = −3
  3. 3. t = 2.071 min
  4. 4. y(1.5) = 0.1389
  5. 5. y(2) = 2.667
  6. 6. y(1) = 1.431

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