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)N17 hN = ?

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.39 h⁻¹. How many cells N are there after 17 h?

N0 = 200k = 0.39 h⁻¹t = 17 h
N =
2.Easy
rr² + br + c?

For the equation y″ + 6y′ + 8y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 8y = 0
r =
3.Medium
tyy(0) = 4y(0.5) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 4 and y′(0) = -2. Find y(0.5).

y″ + 5y′ + 6y = 0y(0) = 4y′(0) = -2
y(0.5) =
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 5 and y′(0) = -2. Find y(2).

y″ + 8y′ + 16y = 0y(0) = 5y′(0) = -2
y(2) =
5.Hard
xy13(1, 4)y(3) = ?

Solve the initial value problem y′ + 2y/x = 6x for x > 0, with y(1) = 4. Find y(3).

y′ + 2y/x = 6xy(1) = 4
y(3) =
6.Hard
tyK = 1y(1.5) = ?

A system obeys y″ + 3y′ + 6.25y = 6.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 3y′ + 6.25y = 6.25y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. N = 151496
  2. 2. r = −2
  3. 3. y(0.5) = 2.34
  4. 4. y(2) = 0.01375
  5. 5. y(3) = 13.78
  6. 6. y(1.5) = 1.093

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