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)y2 h720

A quantity grows according to dy/dt = k·y with k > 0. It is 200 at t = 0 and 720 at t = 2 h. Find the rate constant k in h⁻¹.

y(0) = 200y(2) = 720
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + 2.4y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 16y = 0
ζ =
3.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 3 and y′(0) = 4. Find y(0.5).

y″ + 7y′ + 10y = 0y(0) = 3y′(0) = 4
y(0.5) =
4.Medium
t (min)T (°C)9Ta​69 °C

A cup of coffee at 90 °C is left in a room at 21 °C. After 9 min its temperature is 69 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 90 °CTa = 21 °CT(9) = 69 °C
k = min⁻¹
5.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 3y′ + 18.25y = 182.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 18.25y = 182.5y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 1y(0.25) = ?

A system obeys y″ + 4y′ + 13y = 13 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + 4y′ + 13y = 13y(0) = 0y′(0) = 0
y(0.25) =

Answer key

  1. 1. k = 0.6405 h⁻¹
  2. 2. ζ = 0.3
  3. 3. y(0.5) = 2.056
  4. 4. k = 0.04032 min⁻¹
  5. 5. ymax = 13.08
  6. 6. y(0.25) = 0.2806

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