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″ + 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 =
2.Easy
t (h)y9 h130

A quantity decays according to dy/dt = −k·y with k > 0. It is 200 at t = 0 and 130 at t = 9 h. Find the rate constant k in h⁻¹.

y(0) = 200y(9) = 130
k = h⁻¹
3.Medium
t (min)T (°C)11Ta​52 °C

A cup of coffee at 83 °C is left in a room at 19 °C. After 11 min its temperature is 52 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 83 °CTa = 19 °CT(11) = 52 °C
k = min⁻¹
4.Medium
tyy(0) = 2y(1) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 2 and y′(0) = 6. Find y(1).

y″ + 8y′ + 15y = 0y(0) = 2y′(0) = 6
y(1) =
5.Hard
tyK = 1y(0.5) = ?

A system obeys y″ + y′ + 4.25y = 4.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + y′ + 4.25y = 4.25y(0) = 0y′(0) = 0
y(0.5) =
6.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 2y′ + 17y = 170 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 17y = 170y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. r = −2
  2. 2. k = 0.04786 h⁻¹
  3. 3. k = 0.06022 min⁻¹
  4. 4. y(1) = 0.3579
  5. 5. y(0.5) = 0.4154
  6. 6. ymax = 14.56

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