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)N15 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.37 h⁻¹. How many cells N are there after 15 h?

N0 = 5000k = 0.37 h⁻¹t = 15 h
N =
2.Easy
t (h)N3 hN = ?

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.08 h⁻¹. How many cells N are there after 3 h?

N0 = 2000k = 0.08 h⁻¹t = 3 h
N =
3.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 2 and y′(0) = 4. Find y(0.5).

y″ + 8y′ + 15y = 0y(0) = 2y′(0) = 4
y(0.5) =
4.Medium
t (min)T (°C)14Ta​46 °C

A cup of coffee at 83 °C is left in a room at 23 °C. After 14 min its temperature is 46 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 83 °CTa = 23 °CT(14) = 46 °C
k = min⁻¹
5.Hard
tyK = 2y(1) = ?

A system obeys y″ + 4y′ + 13y = 26 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 4y′ + 13y = 26y(0) = 0y′(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. N = 1.286 × 10⁶
  2. 2. N = 2542
  3. 3. y(0.5) = 1.151
  4. 4. k = 0.06849 min⁻¹
  5. 5. y(1) = 2.242
  6. 6. ymax = 1.043

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