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
t (h)y6 h1160

A quantity grows according to dy/dt = k·y with k > 0. It is 400 at t = 0 and 1160 at t = 6 h. Find the rate constant k in h⁻¹.

y(0) = 400y(6) = 1160
k = h⁻¹
2.Easy
rr² + br + c?

For the equation y″ + 6y′ + 5y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 5y = 0
r =
3.Medium
tyy(0) = 5y(1) = ?

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

y″ + 2y′ + y = 0y(0) = 5y′(0) = 0
y(1) =
4.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 3 and y′(0) = 2. Find y(0.5).

y″ + 8y′ + 16y = 0y(0) = 3y′(0) = 2
y(0.5) =
5.Hard
ty0.5y(0.5) = ?

Solve y′ + 0.5y = 6e−3t with y(0) = 5. Find y(0.5).

y′ + 0.5y = 6e−3ty(0) = 5
y(0.5) =
6.Hard
ty0.5y(0.5) = ?

Solve y′ + 3y = 10e−2t with y(0) = 5. Find y(0.5).

y′ + 3y = 10e−2ty(0) = 5
y(0.5) =

Answer key

  1. 1. k = 0.1775 h⁻¹
  2. 2. r = −1
  3. 3. y(1) = 3.679
  4. 4. y(0.5) = 1.353
  5. 5. y(0.5) = 5.228
  6. 6. y(0.5) = 2.563

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