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)6 dm = ?

A radioactive tracer has a half-life of 25 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 6 days?

m0 = 10 mgt½ = 25 dt = 6 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 9y′ + 18y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 9y′ + 18y = 0
r =
3.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 1 and y′(0) = 4. Find y(0.5).

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

Solve y″ + 6y′ + 8y = 0 with y(0) = 1 and y′(0) = -2. Find y(1).

y″ + 6y′ + 8y = 0y(0) = 1y′(0) = -2
y(1) =
5.Hard
tyK = 2y(1.5) = ?

A system obeys y″ + 3y′ + 11.25y = 22.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 3y′ + 11.25y = 22.5y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
tyy(0) = 2y(0.5) = ?

Solve y″ + 2y′ + 10y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 2y′ + 10y = 0y(0) = 2y′(0) = 0
y(0.5) =

Answer key

  1. 1. m = 8.467 mg
  2. 2. r = −3
  3. 3. y(0.5) = 1.004
  4. 4. y(1) = 0.1353
  5. 5. y(1.5) = 2.147
  6. 6. y(0.5) = 0.4891

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