First-Order Linear Equations
Use an integrating factor.
Key idea
Use an integrating factor.
In depth
y′ + P(x)y = Q(x) is solved with the integrating factor μ = e^(∫P dx). Multiplying turns the left side into (μy)′, so integrate and divide by μ.
Worked example
y′ + 2y = e^x: μ = e^(2x), (e^(2x)y)′ = e^(3x), y = e^x/3 + Ce^(−2x).
Formula
Real-world uses
- Modelling loans with repayments, leaking tanks with inflow and RC charging circuits
Common mistake
Forgetting to divide by μ at the end, or not writing the equation in standard form with coefficient 1 on y′.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is First-Order Linear Equations?
Use an integrating factor. It is a topic at the advanced level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for First-Order Linear Equations?
The key formula is y′ + p(x)y = q(x).
Where is First-Order Linear Equations used in real life?
Real-world examples: Modelling loans with repayments, leaking tanks with inflow and RC charging circuits.
What should I learn before First-Order Linear Equations?
Make sure you are comfortable with Separable Equations, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with First-Order Linear Equations?
Forgetting to divide by μ at the end, or not writing the equation in standard form with coefficient 1 on y′.
How can I practise First-Order Linear Equations?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after First-Order Linear Equations?
Move on to Exponential Growth and Decay Models next.
Related advanced topics in Differential Equations, Abstract Algebra & Applied Math
- What Is a Differential Equation? – An equation relating a function and its derivatives.
- Order and Linearity of ODEs – Classify equations by highest derivative and linearity.
- Direction Fields – Slope fields show solution curves visually.
- Separable Equations – Separate variables and integrate both sides.
- Exponential Growth and Decay Models – dy/dt = ky gives y = y₀eᵏᵗ.
- Logistic Growth – Growth limited by carrying capacity.