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)y9 h380

A quantity grows according to dy/dt = k·y with k > 0. It is 200 at t = 0 and 380 at t = 9 h. Find the rate constant k in h⁻¹.

y(0) = 200y(9) = 380
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + 2y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2y′ + 100y = 0
ζ =
3.Medium
t (min)T (°C)24Ta​T = ?

A can of soda at 6 °C is taken out of the fridge into a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.1 min⁻¹, find its temperature T after 24 min.

T0 = 6 °CTa = 22 °Ck = 0.1 min⁻¹t = 24 min
T = °C
4.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 1 and y′(0) = 2. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 1y′(0) = 2
y(1.5) =
5.Hard
ty1.5y(1.5) = ?

Solve y′ + y = 10e−0.5t with y(0) = 2. Find y(1.5).

y′ + y = 10e−0.5ty(0) = 2
y(1.5) =
6.Hard
tyK = 2y(0.25) = ?

A system obeys y″ + y′ + 16.25y = 32.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + y′ + 16.25y = 32.5y(0) = 0y′(0) = 0
y(0.25) =

Answer key

  1. 1. k = 0.07132 h⁻¹
  2. 2. ζ = 0.1
  3. 3. T = 20.55 °C
  4. 4. y(1.5) = 1.227
  5. 5. y(1.5) = 5.431
  6. 6. y(0.25) = 0.8607

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