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″ + 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 =
2.Easy
rr² + br + c?

For the equation y″ + 8y′ + 15y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 15y = 0
r =
3.Medium
t (min)T (°C)19Ta​40 °C31 °C

A cup of coffee at 94 °C is left in a room at 20 °C. After 19 min its temperature is 40 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 31 °C?

T0 = 94 °CTa = 20 °CT(19) = 40 °CT = 31 °C
t = min
4.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 5y′ + 4y = 0 with y(0) = 1 and y′(0) = -4. Find y(0.5).

y″ + 5y′ + 4y = 0y(0) = 1y′(0) = -4
y(0.5) =
5.Hard
tyy(0) = 5y(0.5) = ?

Solve y″ + 3y′ + 18.25y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 3y′ + 18.25y = 0y(0) = 5y′(0) = 0
y(0.5) =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + y′ + 16.25y = 32.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 16.25y = 32.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. r = −4
  2. 2. r = −3
  3. 3. t = 27.68 min
  4. 4. y(0.5) = 0.1353
  5. 5. y(0.5) = −0.1775
  6. 6. ymax = 3.35

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