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)N4 hN = ?

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.17 h⁻¹. How many cells N are there after 4 h?

N0 = 1500k = 0.17 h⁻¹t = 4 h
N =
2.Easy
t (h)N12 hN = ?

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.28 h⁻¹. How many cells N are there after 12 h?

N0 = 2000k = 0.28 h⁻¹t = 12 h
N =
3.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 3 and y′(0) = -1. Find y(0.5).

y″ + 6y′ + 9y = 0y(0) = 3y′(0) = -1
y(0.5) =
4.Medium
t (min)T (°C)19Ta​T = ?

A can of soda at 5 °C is taken out of the fridge into a room at 30 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.09 min⁻¹, find its temperature T after 19 min.

T0 = 5 °CTa = 30 °Ck = 0.09 min⁻¹t = 19 min
T = °C
5.Hard
tyK = 2y(0.25) = ?

A system obeys y″ + 3y′ + 27.25y = 54.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + 3y′ + 27.25y = 54.5y(0) = 0y′(0) = 0
y(0.25) =
6.Hard
tyy(0) = 3y(1) = ?

Solve y″ + 2y′ + 10y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 2y′ + 10y = 0y(0) = 3y′(0) = 0
y(1) =

Answer key

  1. 1. N = 2961
  2. 2. N = 57578
  3. 3. y(0.5) = 1.562
  4. 4. T = 25.48 °C
  5. 5. y(0.25) = 1.175
  6. 6. y(1) = −1.041

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