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″ + 1.6y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.6y′ + 4y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 3y′ + 2y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 3y′ + 2y = 0
r =
3.Medium
tyy(0) = 2y(2) = ?

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

y″ + 6y′ + 9y = 0y(0) = 2y′(0) = 2
y(2) =
4.Medium
t (min)T (°C)15Ta​12 °C10 °C

A can of soda at 5 °C is taken out of the fridge into a room at 24 °C. After 15 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 10 °C?

T0 = 5 °CTa = 24 °CT(15) = 12 °CT = 10 °C
t = min
5.Hard
tyK = 5y(0.5) = ?

A system obeys y″ + 4y′ + 29y = 145 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 4y′ + 29y = 145y(0) = 0y′(0) = 0
y(0.5) =
6.Hard
xy12(1, 0)y(2) = ?

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

y′ + y/x = 5xy(1) = 0
y(2) =

Answer key

  1. 1. ζ = 0.4
  2. 2. r = −1
  3. 3. y(2) = 0.04462
  4. 4. t = 9.968 min
  5. 5. y(0.5) = 6.033
  6. 6. y(2) = 5.833

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