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
t (h)Nt = ?15000

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.28 h⁻¹. After how many hours t are there 15000 cells?

N0 = 1500k = 0.28 h⁻¹N = 15000
t = h
2.Easy
t (h)Nt = ?4500

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.21 h⁻¹. After how many hours t are there 4500 cells?

N0 = 1500k = 0.21 h⁻¹N = 4500
t = h
3.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 1 and y′(0) = 0. Find y(0.5).

y″ + 4y′ + 4y = 0y(0) = 1y′(0) = 0
y(0.5) =
4.Medium
tyy(0) = 5y(1.5) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 5 and y′(0) = 6. Find y(1.5).

y″ + 6y′ + 8y = 0y(0) = 5y′(0) = 6
y(1.5) =
5.Hard
tyy(0) = 10y(1.5) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 10y′(0) = 0
y(1.5) =
6.Hard
tyK = 5y(0.75) = ?

A system obeys y″ + 3y′ + 11.25y = 56.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 3y′ + 11.25y = 56.25y(0) = 0y′(0) = 0
y(0.75) =

Answer key

  1. 1. t = 8.224 h
  2. 2. t = 5.231 h
  3. 3. y(0.5) = 0.7358
  4. 4. y(1.5) = 0.6274
  5. 5. y(1.5) = 1.192
  6. 6. y(0.75) = 5.388

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