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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.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 =
2.Easy
t (days)m (mg)38 dm = ?

A radioactive tracer has a half-life of 17 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 38 days?

m0 = 120 mgt½ = 17 dt = 38 d
m = mg
3.Medium
tyy(0) = 1y(1) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 1 and y′(0) = -2. Find y(1).

y″ + 8y′ + 16y = 0y(0) = 1y′(0) = -2
y(1) =
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 5 and y′(0) = -1. Find y(2).

y″ + 4y′ + 4y = 0y(0) = 5y′(0) = -1
y(2) =
5.Hard
tyK = 1y(0.75) = ?

A system obeys y″ + 4y′ + 20y = 20 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 4y′ + 20y = 20y(0) = 0y′(0) = 0
y(0.75) =
6.Hard
tyy(0) = 5y(0.25) = ?

Solve y″ + 4y′ + 8y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 5y′(0) = 0
y(0.25) =

Answer key

  1. 1. r = −1
  2. 2. m = 25.49 mg
  3. 3. y(1) = 0.05495
  4. 4. y(2) = 0.4213
  5. 5. y(0.75) = 1.205
  6. 6. y(0.25) = 4.115

Full step-by-step solutions: practise this topic on examproblems.com.