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″ + 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 =
2.Easy
ty1

A mass–spring–damper is described by y″ + 10y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 10y′ + 100y = 0
ζ =
3.Medium
t (min)T (°C)5Ta​12 °C17 °C

A can of soda at 4 °C is taken out of the fridge into a room at 24 °C. After 5 min its temperature is 12 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 17 °C?

T0 = 4 °CTa = 24 °CT(5) = 12 °CT = 17 °C
t = min
4.Medium
tyy(0) = 3y(1) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 3 and y′(0) = 0. Find y(1).

y″ + 6y′ + 8y = 0y(0) = 3y′(0) = 0
y(1) =
5.Hard
xy12(1, 1)y(2) = ?

Solve the initial value problem y′ + 3y/x = 5 for x > 0, with y(1) = 1. Find y(2).

y′ + 3y/x = 5y(1) = 1
y(2) =
6.Hard
tyK = 1y(1) = ?

A system obeys y″ + 3y′ + 6.25y = 6.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 3y′ + 6.25y = 6.25y(0) = 0y′(0) = 0
y(1) =

Answer key

  1. 1. r = −3
  2. 2. ζ = 0.5
  3. 3. t = 10.28 min
  4. 4. y(1) = 0.7571
  5. 5. y(2) = 2.469
  6. 6. y(1) = 0.9407

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