Separable Equations
Separate variables and integrate both sides.
Key idea
Separate variables and integrate both sides.
In depth
If y′ = g(x)h(y), separate variables: dy/h(y) = g(x) dx and integrate both sides. It works because each side then depends on one variable only.
Worked example
y′ = xy: dy/y = x dx → ln|y| = x²/2 + C → y = Ke^(x²/2).
Formula
Real-world uses
- Predicting how long a cup of coffee takes to cool or how a tank drains through a hole
Common mistake
Dividing by h(y) and losing constant solutions where h(y) = 0.
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 Separable Equations?
Separate variables and integrate both sides. It is a topic at the advanced level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for Separable Equations?
The key formula is ∫dy/g(y) = ∫f(x)dx.
Where is Separable Equations used in real life?
Real-world examples: Predicting how long a cup of coffee takes to cool or how a tank drains through a hole.
What should I learn before Separable Equations?
Make sure you are comfortable with Direction Fields, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Separable Equations?
Dividing by h(y) and losing constant solutions where h(y) = 0.
How can I practise Separable 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 Separable Equations?
Move on to First-Order Linear Equations 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.
- First-Order Linear Equations – Use an integrating factor.
- Exponential Growth and Decay Models – dy/dt = ky gives y = y₀eᵏᵗ.
- Logistic Growth – Growth limited by carrying capacity.