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)66 dm = ?

A radioactive tracer has a half-life of 21 days. A sample initially contains 80 mg. The decay follows dm/dt = −k·m. How much m remains after 66 days?

m0 = 80 mgt½ = 21 dt = 66 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 12y = 0
r =
3.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = 4. Find y(1).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = 4
y(1) =
4.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 3y′ + 2y = 0 with y(0) = 5 and y′(0) = -4. Find y(0.5).

y″ + 3y′ + 2y = 0y(0) = 5y′(0) = -4
y(0.5) =
5.Hard
tyK = 10y(0.75) = ?

A system obeys y″ + y′ + 16.25y = 162.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + y′ + 16.25y = 162.5y(0) = 0y′(0) = 0
y(0.75) =
6.Hard
tyK = 1y(0.25) = ?

A system obeys y″ + 4y′ + 29y = 29 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + 4y′ + 29y = 29y(0) = 0y′(0) = 0
y(0.25) =

Answer key

  1. 1. m = 9.057 mg
  2. 2. r = −3
  3. 3. y(1) = 1.195
  4. 4. y(0.5) = 3.271
  5. 5. y(0.75) = 16.68
  6. 6. y(0.25) = 0.5785

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