LaTeX
y=e^{-\int P(x)dx}\left(\int Q(x)e^{\int P(x)dx}dx+C\right)Variables
- P(x): coefficient function
- Q(x): forcing function
- C: integration constant
How to use this formula
Gives the integrating-factor solution of y′+P(x)y=Q(x).
Important notes
- Continuity assumptions are needed on the interval of interest.
- Apply initial data after finding the general solution.
Quick example
For y′+y=1, the solution is y=1+Ce^{-x}.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Continuity assumptions are needed on the interval of interest.
- Integration limits, the differential, and any constant of integration must be included where required.
- State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method.
Worked example
For y′+y=1, the solution is y=1+Ce^{-x}.
Common mistakes
- Keep the integration bounds and differential attached to First-Order Linear ODE Solution, and include a constant of integration for an indefinite integral.
- Do not present a general solution as the requested solution before applying all initial or boundary conditions.
Continue the workflow
Use First-Order Linear ODE Solution in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
y=e^{-\int P(x)dx}\left(\int Q(x)e^{\int P(x)dx}dx+C\right). - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the First-Order Linear ODE Solution used for?
Gives the integrating-factor solution of y′+P(x)y=Q(x).
Can I copy this formula as LaTeX?
Yes. Copy y=e^{-\int P(x)dx}\left(\int Q(x)e^{\int P(x)dx}dx+C\right) or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.