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)y7 h390

A quantity grows according to dy/dt = k·y with k > 0. It is 100 at t = 0 and 390 at t = 7 h. Find the rate constant k in h⁻¹.

y(0) = 100y(7) = 390
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + 3.6y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3.6y′ + 9y = 0
ζ =
3.Medium
t (min)T (°C)5Ta​T = ?

A cup of coffee at 81 °C is left in a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.06 min⁻¹, find its temperature T after 5 min.

T0 = 81 °CTa = 22 °Ck = 0.06 min⁻¹t = 5 min
T = °C
4.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 2 and y′(0) = 4. Find y(1.5).

y″ + 6y′ + 9y = 0y(0) = 2y′(0) = 4
y(1.5) =
5.Hard
xy12(1, 5)y(2) = ?

Solve the initial value problem y′ + 2y/x = 2 for x > 0, with y(1) = 5. Find y(2).

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

A system obeys y″ + 2y′ + 10y = 10 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 10y = 10y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. k = 0.1944 h⁻¹
  2. 2. ζ = 0.6
  3. 3. T = 65.71 °C
  4. 4. y(1.5) = 0.1889
  5. 5. y(2) = 2.417
  6. 6. ymax = 1.351

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