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″ + 1.5y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.5y′ + 9y = 0
ζ =
2.Easy
t (days)m (mg)1 dm = ?

A radioactive tracer has a half-life of 21 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 1 days?

m0 = 120 mgt½ = 21 dt = 1 d
m = mg
3.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 2 and y′(0) = 0. Find y(0.5).

y″ + 6y′ + 9y = 0y(0) = 2y′(0) = 0
y(0.5) =
4.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 5 and y′(0) = 1. Find y(0.5).

y″ + 4y′ + 4y = 0y(0) = 5y′(0) = 1
y(0.5) =
5.Hard
xy12(1, 1)y(2) = ?

Solve the initial value problem y′ + 3y/x = 6 for x > 0, with y(1) = 1. Find y(2).

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

A system obeys y″ + y′ + 16.25y = 16.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 16.25y = 16.25y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.25
  2. 2. m = 116.1 mg
  3. 3. y(0.5) = 1.116
  4. 4. y(0.5) = 3.863
  5. 5. y(2) = 2.938
  6. 6. ymax = 1.675

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