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)y11 h290

A quantity grows according to dy/dt = k·y with k > 0. It is 100 at t = 0 and 290 at t = 11 h. Find the rate constant k in h⁻¹.

y(0) = 100y(11) = 290
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + 1.6y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.6y′ + 4y = 0
ζ =
3.Medium
t (min)T (°C)34Ta​176 °C

A forged steel part at 440 °C is left to cool in a workshop at 27 °C. After 34 min its temperature is 176 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 440 °CTa = 27 °CT(34) = 176 °C
k = min⁻¹
4.Medium
t (min)T (°C)40Ta​86 °C

A forged steel part at 460 °C is left to cool in a workshop at 28 °C. After 40 min its temperature is 86 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 460 °CTa = 28 °CT(40) = 86 °C
k = min⁻¹
5.Hard
xy13(1, 5)y(3) = ?

Solve the initial value problem y′ + 3y/x = 6 for x > 0, with y(1) = 5. Find y(3).

y′ + 3y/x = 6y(1) = 5
y(3) =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 4y′ + 20y = 40 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 20y = 40y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. k = 0.09679 h⁻¹
  2. 2. ζ = 0.4
  3. 3. k = 0.02999 min⁻¹
  4. 4. k = 0.0502 min⁻¹
  5. 5. y(3) = 4.63
  6. 6. ymax = 2.416

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