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″ + 4.8y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 9y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 3y′ + 2y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 3y′ + 2y = 0
r =
3.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = 2. Find y(2).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = 2
y(2) =
4.Medium
t (min)T (°C)20Ta​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 20 min.

T0 = 6 °CTa = 22 °Ck = 0.1 min⁻¹t = 20 min
T = °C
5.Hard
tyy(0) = 5y(0.5) = ?

Solve y″ + 4y′ + 29y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 4y′ + 29y = 0y(0) = 5y′(0) = 0
y(0.5) =
6.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 2y′ + 26y = 130 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 26y = 130y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.8
  2. 2. r = −1
  3. 3. y(2) = 0.09667
  4. 4. T = 19.83 °C
  5. 5. y(0.5) = −1.033
  6. 6. ymax = 7.667

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