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)y5 h63

A quantity decays according to dy/dt = −k·y with k > 0. It is 250 at t = 0 and 63 at t = 5 h. Find the rate constant k in h⁻¹.

y(0) = 250y(5) = 63
k = h⁻¹
2.Easy
t (h)N4 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.26 h⁻¹. How many cells N are there after 4 h?

N0 = 500k = 0.26 h⁻¹t = 4 h
N =
3.Medium
t (min)T (°C)24Ta​T = ?

A cup of coffee at 92 °C is left in a room at 21 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.06 min⁻¹, find its temperature T after 24 min.

T0 = 92 °CTa = 21 °Ck = 0.06 min⁻¹t = 24 min
T = °C
4.Medium
t (min)T (°C)25Ta​19 °C

A can of soda at 3 °C is taken out of the fridge into a room at 22 °C. After 25 min its temperature is 19 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 3 °CTa = 22 °CT(25) = 19 °C
k = min⁻¹
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 2y′ + 26y = 26 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

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

A system obeys y″ + 3y′ + 18.25y = 36.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 3y′ + 18.25y = 36.5y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. k = 0.2757 h⁻¹
  2. 2. N = 1415
  3. 3. T = 37.82 °C
  4. 4. k = 0.07383 min⁻¹
  5. 5. ymax = 1.533
  6. 6. y(1.5) = 1.82

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