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″ + y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + y′ + 4y = 0
ζ =
2.Easy
t (h)y7 h775

A quantity grows according to dy/dt = k·y with k > 0. It is 250 at t = 0 and 775 at t = 7 h. Find the rate constant k in h⁻¹.

y(0) = 250y(7) = 775
k = h⁻¹
3.Medium
t (min)T (°C)17Ta​36 °C57 °C

A cup of coffee at 89 °C is left in a room at 21 °C. After 17 min its temperature is 36 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 57 °C?

T0 = 89 °CTa = 21 °CT(17) = 36 °CT = 57 °C
t = min
4.Medium
t (min)T (°C)27Ta​T = ?

A can of soda at 3 °C is taken out of the fridge into a room at 25 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 27 min.

T0 = 3 °CTa = 25 °Ck = 0.04 min⁻¹t = 27 min
T = °C
5.Hard
tyy(0) = 10y(1.5) = ?

Solve y″ + 3y′ + 18.25y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 3y′ + 18.25y = 0y(0) = 10y′(0) = 0
y(1.5) =
6.Hard
ty2y(2) = ?

Solve y′ + 3y = 9e−0.5t with y(0) = 1. Find y(2).

y′ + 3y = 9e−0.5ty(0) = 1
y(2) =

Answer key

  1. 1. ζ = 0.25
  2. 2. k = 0.1616 h⁻¹
  3. 3. t = 7.153 min
  4. 4. T = 17.53 °C
  5. 5. y(1.5) = 0.9016
  6. 6. y(2) = 1.318

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