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″ + 18y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 18y′ + 36y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 4y′ + 3y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 4y′ + 3y = 0
r =
3.Medium
tyy(0) = 2y(1.5) = ?

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

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

Solve y″ + 4y′ + 4y = 0 with y(0) = 3 and y′(0) = 2. Find y(1.5).

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = 2
y(1.5) =
5.Hard
ty0.5y(0.5) = ?

Solve y′ + 3y = 6e−0.5t with y(0) = 1. Find y(0.5).

y′ + 3y = 6e−0.5ty(0) = 1
y(0.5) =
6.Hard
tyK = 10y(1.5) = ?

A system obeys y″ + y′ + 4.25y = 42.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + y′ + 4.25y = 42.5y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. ζ = 1.5
  2. 2. r = −1
  3. 3. y(1.5) = 0.697
  4. 4. y(1.5) = 0.7468
  5. 5. y(0.5) = 1.557
  6. 6. y(1.5) = 14.51

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