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 (days)m (mg)83 dm = ?

A radioactive tracer has a half-life of 26 days. A sample initially contains 40 mg. The decay follows dm/dt = −k·m. How much m remains after 83 days?

m0 = 40 mgt½ = 26 dt = 83 d
m = mg
2.Easy
ty1

A mass–spring–damper is described by y″ + 8y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 8y′ + 25y = 0
ζ =
3.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 3 and y′(0) = 1. Find y(1.5).

y″ + 6y′ + 8y = 0y(0) = 3y′(0) = 1
y(1.5) =
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = 1. Find y(2).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = 1
y(2) =
5.Hard
tyK = 10y(1) = ?

A system obeys y″ + 4y′ + 8y = 80 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 4y′ + 8y = 80y(0) = 0y′(0) = 0
y(1) =
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. m = 4.376 mg
  2. 2. ζ = 0.8
  3. 3. y(1.5) = 0.3149
  4. 4. y(2) = 0.09171
  5. 5. y(1) = 9.333
  6. 6. ymax = 1.351

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