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)y2 h60

A quantity decays according to dy/dt = −k·y with k > 0. It is 80 at t = 0 and 60 at t = 2 h. Find the rate constant k in h⁻¹.

y(0) = 80y(2) = 60
k = h⁻¹
2.Easy
t (h)y9 h228

A quantity grows according to dy/dt = k·y with k > 0. It is 120 at t = 0 and 228 at t = 9 h. Find the rate constant k in h⁻¹.

y(0) = 120y(9) = 228
k = h⁻¹
3.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 4y′ + 3y = 0 with y(0) = 2 and y′(0) = -4. Find y(1.5).

y″ + 4y′ + 3y = 0y(0) = 2y′(0) = -4
y(1.5) =
4.Medium
t (min)T (°C)13Ta​15 °C

A can of soda at 5 °C is taken out of the fridge into a room at 25 °C. After 13 min its temperature is 15 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 5 °CTa = 25 °CT(13) = 15 °C
k = min⁻¹
5.Hard
tyK = 2y(1.5) = ?

A system obeys y″ + 2y′ + 26y = 52 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 2y′ + 26y = 52y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 4y′ + 13y = 26 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 13y = 26y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. k = 0.1438 h⁻¹
  2. 2. k = 0.07132 h⁻¹
  3. 3. y(1.5) = 0.2342
  4. 4. k = 0.05332 min⁻¹
  5. 5. y(1.5) = 1.762
  6. 6. ymax = 2.246

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