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 = ?6000

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.14 h⁻¹. After how many hours t are there 6000 cells?

N0 = 2000k = 0.14 h⁻¹N = 6000
t = h
2.Easy
rr² + br + c?

For the equation y″ + 9y′ + 20y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 9y′ + 20y = 0
r =
3.Medium
tyy(0) = 4y(1) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 4 and y′(0) = -4. Find y(1).

y″ + 8y′ + 15y = 0y(0) = 4y′(0) = -4
y(1) =
4.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 3 and y′(0) = 0. Find y(2).

y″ + 6y′ + 9y = 0y(0) = 3y′(0) = 0
y(2) =
5.Hard
tyK = 5y(0.25) = ?

A system obeys y″ + 2y′ + 17y = 85 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + 2y′ + 17y = 85y(0) = 0y′(0) = 0
y(0.25) =
6.Hard
tyK = 10ymax​ = ?

A system obeys y″ + y′ + 25.25y = 252.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 25.25y = 252.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. t = 7.847 h
  2. 2. r = −4
  3. 3. y(1) = 0.3713
  4. 4. y(2) = 0.05205
  5. 5. y(0.25) = 2.077
  6. 6. ymax = 17.3

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