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 worksheet Practise 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

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.35 h⁻¹. After how many hours t are there 10000 cells?

N₀ = 2000k = 0.35 h⁻¹N = 10000

t = h

2.Easy

A quantity decays according to dy/dt = −k·y with k > 0. It is 250 at t = 0 and 213 at t = 2 h. Find the rate constant k in h⁻¹.

y(0) = 250y(2) = 213

k = h⁻¹

3.Medium

A can of soda at 5 °C is taken out of the fridge into a room at 28 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 13 min.

T₀ = 5 °CT_a = 28 °Ck = 0.04 min⁻¹t = 13 min

T = °C

4.Medium

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

y″ + 2y′ + y = 0y(0) = 5y′(0) = 1

y(1) =

5.Hard

A system obeys y″ + 3y′ + 6.25y = 31.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 3y′ + 6.25y = 31.25y(0) = 0y′(0) = 0

y(0.5) =

6.Hard

A system obeys y″ + y′ + 4.25y = 8.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + y′ + 4.25y = 8.5y(0) = 0y′(0) = 0

y(0.25) =

Answer key

1. t = 4.598 h
2. k = 0.08008 h⁻¹
3. T = 14.33 °C
4. y(1) = 4.047
5. y(0.5) = 2.233
6. y(0.25) = 0.2395

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