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)y12 h180

A quantity grows according to dy/dt = k·y with k > 0. It is 50 at t = 0 and 180 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 50y(12) = 180
k = h⁻¹
2.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 =
3.Medium
t (min)T (°C)10Ta​47 °C29 °C

A cup of coffee at 82 °C is left in a room at 19 °C. After 10 min its temperature is 47 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 29 °C?

T0 = 82 °CTa = 19 °CT(10) = 47 °CT = 29 °C
t = min
4.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 5 and y′(0) = 0. Find y(0.5).

y″ + 6y′ + 8y = 0y(0) = 5y′(0) = 0
y(0.5) =
5.Hard
tyy(0) = 3y(1) = ?

Solve y″ + y′ + 16.25y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + y′ + 16.25y = 0y(0) = 3y′(0) = 0
y(1) =
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 4y′ + 8y = 8 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 8y = 8y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. k = 0.1067 h⁻¹
  2. 2. r = −3
  3. 3. t = 22.7 min
  4. 4. y(0.5) = 3.002
  5. 5. y(1) = −1.361
  6. 6. ymax = 1.043

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