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″ + 0.8y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 0.8y′ + 16y = 0
ζ =
2.Easy
t (days)m (mg)37 dm = ?

A radioactive tracer has a half-life of 22 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 37 days?

m0 = 10 mgt½ = 22 dt = 37 d
m = mg
3.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 2 and y′(0) = 6. Find y(1.5).

y″ + 8y′ + 15y = 0y(0) = 2y′(0) = 6
y(1.5) =
4.Medium
tyy(0) = 1y(2) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 1 and y′(0) = 0. Find y(2).

y″ + 4y′ + 4y = 0y(0) = 1y′(0) = 0
y(2) =
5.Hard
ty1y(1) = ?

Solve y′ + 0.5y = 3e−2t with y(0) = 5. Find y(1).

y′ + 0.5y = 3e−2ty(0) = 5
y(1) =
6.Hard
xy13(1, 0)y(3) = ?

Solve the initial value problem y′ + 3y/x = 3x2 for x > 0, with y(1) = 0. Find y(3).

y′ + 3y/x = 3x2y(1) = 0
y(3) =

Answer key

  1. 1. ζ = 0.1
  2. 2. m = 3.117 mg
  3. 3. y(1.5) = 0.08555
  4. 4. y(2) = 0.09158
  5. 5. y(1) = 3.975
  6. 6. y(3) = 13.48

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