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
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 12y = 0
r =
2.Easy
t (h)N8 hN = ?

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.07 h⁻¹. How many cells N are there after 8 h?

N0 = 1000k = 0.07 h⁻¹t = 8 h
N =
3.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 2 and y′(0) = -2. Find y(1.5).

y″ + 4y′ + 4y = 0y(0) = 2y′(0) = -2
y(1.5) =
4.Medium
t (min)T (°C)16Ta​53 °C62 °C

A cup of coffee at 83 °C is left in a room at 20 °C. After 16 min its temperature is 53 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 62 °C?

T0 = 83 °CTa = 20 °CT(16) = 53 °CT = 62 °C
t = min
5.Hard
tyy(0) = 2y(0.5) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 2y′(0) = 0
y(0.5) =
6.Hard
tyK = 5y(1) = ?

A system obeys y″ + 3y′ + 11.25y = 56.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 3y′ + 11.25y = 56.25y(0) = 0y′(0) = 0
y(1) =

Answer key

  1. 1. r = −3
  2. 2. N = 1751
  3. 3. y(1.5) = 0.2489
  4. 4. t = 10.03 min
  5. 5. y(0.5) = −0.8266
  6. 6. y(1) = 6.026

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