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″ + 1.5y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.5y′ + 9y = 0
ζ =
2.Easy
t (h)Nt = ?50000

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.3 h⁻¹. After how many hours t are there 50000 cells?

N0 = 5000k = 0.3 h⁻¹N = 50000
t = h
3.Medium
tyy(0) = 2y(1.5) = ?

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

y″ + 2y′ + y = 0y(0) = 2y′(0) = 2
y(1.5) =
4.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 3 and y′(0) = 4. Find y(2).

y″ + 2y′ + y = 0y(0) = 3y′(0) = 4
y(2) =
5.Hard
tyy(0) = 3y(0.5) = ?

Solve y″ + 4y′ + 29y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 4y′ + 29y = 0y(0) = 3y′(0) = 0
y(0.5) =
6.Hard
tyK = 5y(1.5) = ?

A system obeys y″ + 2y′ + 17y = 85 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 2y′ + 17y = 85y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. ζ = 0.25
  2. 2. t = 7.675 h
  3. 3. y(1.5) = 1.785
  4. 4. y(2) = 2.301
  5. 5. y(0.5) = −0.62
  6. 6. y(1.5) = 4.007

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