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
ty1

A mass–spring–damper is described by y″ + 2.4y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 36y = 0
ζ =
2.Easy
t (h)N20 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.06 h⁻¹. How many cells N are there after 20 h?

N0 = 500k = 0.06 h⁻¹t = 20 h
N =
3.Medium
tyy(0) = 2y(2) = ?

Solve y″ + 3y′ + 2y = 0 with y(0) = 2 and y′(0) = 1. Find y(2).

y″ + 3y′ + 2y = 0y(0) = 2y′(0) = 1
y(2) =
4.Medium
t (min)T (°C)21Ta​26 °C

A cup of coffee at 94 °C is left in a room at 19 °C. After 21 min its temperature is 26 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 94 °CTa = 19 °CT(21) = 26 °C
k = min⁻¹
5.Hard
tyy(0) = 5y(0.25) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 5y′(0) = 0
y(0.25) =
6.Hard
tyK = 10y(1.5) = ?

A system obeys y″ + 4y′ + 8y = 80 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 4y′ + 8y = 80y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. ζ = 0.2
  2. 2. N = 1660
  3. 3. y(2) = 0.6217
  4. 4. k = 0.1129 min⁻¹
  5. 5. y(0.25) = 1.967
  6. 6. y(1.5) = 10.42

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