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 = ?10000

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.09 h⁻¹. After how many hours t are there 10000 cells?

N0 = 1000k = 0.09 h⁻¹N = 10000
t = h
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
t (min)T (°C)19Ta​17 °C12 °C

A can of soda at 6 °C is taken out of the fridge into a room at 26 °C. After 19 min its temperature is 17 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 12 °C?

T0 = 6 °CTa = 26 °CT(19) = 17 °CT = 12 °C
t = min
4.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 1 and y′(0) = 0. Find y(0.5).

y″ + 8y′ + 15y = 0y(0) = 1y′(0) = 0
y(0.5) =
5.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 4y′ + 13y = 65 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 13y = 65y(0) = 0y′(0) = 0
ymax =
6.Hard
xy12(1, 3)y(2) = ?

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

y′ + 2y/x = 2xy(1) = 3
y(2) =

Answer key

  1. 1. t = 25.58 h
  2. 2. r = −1
  3. 3. t = 8.487 min
  4. 4. y(0.5) = 0.4347
  5. 5. ymax = 5.616
  6. 6. y(2) = 2.625

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