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)N10 hN = ?

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.35 h⁻¹. How many cells N are there after 10 h?

N0 = 2000k = 0.35 h⁻¹t = 10 h
N =
2.Easy
t (h)N19 hN = ?

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.11 h⁻¹. How many cells N are there after 19 h?

N0 = 200k = 0.11 h⁻¹t = 19 h
N =
3.Medium
t (min)T (°C)19Ta​418 °C

A forged steel part at 500 °C is left to cool in a workshop at 28 °C. After 19 min its temperature is 418 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 500 °CTa = 28 °CT(19) = 418 °C
k = min⁻¹
4.Medium
t (min)T (°C)30Ta​230 °C

A forged steel part at 300 °C is left to cool in a workshop at 29 °C. After 30 min its temperature is 230 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 300 °CTa = 29 °CT(30) = 230 °C
k = min⁻¹
5.Hard
xy13(1, −1)y(3) = ?

Solve the initial value problem y′ + y/x = 4x for x > 0, with y(1) = −1. Find y(3).

y′ + y/x = 4xy(1) = −1
y(3) =
6.Hard
tyy(0) = 2y(0.5) = ?

Solve y″ + y′ + 9.25y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + y′ + 9.25y = 0y(0) = 2y′(0) = 0
y(0.5) =

Answer key

  1. 1. N = 66231
  2. 2. N = 1617
  3. 3. k = 0.01004 min⁻¹
  4. 4. k = 0.00996 min⁻¹
  5. 5. y(3) = 11.22
  6. 6. y(0.5) = 0.3691

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