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)N14 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.27 h⁻¹. How many cells N are there after 14 h?

N0 = 500k = 0.27 h⁻¹t = 14 h
N =
2.Easy
t (h)y11 h80

A quantity decays according to dy/dt = −k·y with k > 0. It is 100 at t = 0 and 80 at t = 11 h. Find the rate constant k in h⁻¹.

y(0) = 100y(11) = 80
k = h⁻¹
3.Medium
t (min)T (°C)19Ta​T = ?

A cup of coffee at 83 °C is left in a room at 19 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.11 min⁻¹, find its temperature T after 19 min.

T0 = 83 °CTa = 19 °Ck = 0.11 min⁻¹t = 19 min
T = °C
4.Medium
t (min)T (°C)44Ta​145 °C

A forged steel part at 480 °C is left to cool in a workshop at 23 °C. After 44 min its temperature is 145 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 480 °CTa = 23 °CT(44) = 145 °C
k = min⁻¹
5.Hard
tyK = 10y(1.5) = ?

A system obeys y″ + 2y′ + 10y = 100 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 2y′ + 10y = 100y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
tyy(0) = 1y(1) = ?

Solve y″ + y′ + 9.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + y′ + 9.25y = 0y(0) = 1y′(0) = 0
y(1) =

Answer key

  1. 1. N = 21908
  2. 2. k = 0.02029 h⁻¹
  3. 3. T = 26.92 °C
  4. 4. k = 0.03002 min⁻¹
  5. 5. y(1.5) = 11.2
  6. 6. y(1) = −0.5862

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