Separable Equations

Level 1 - Math II (Physics) topic page in Ordinary Differential Equations.

Principle

Separable Equations is about solving first-order equations by moving all y-dependence to one side and x-dependence to the other. The page treats the idea as a local tool: identify the variables, state the assumptions, then apply the relevant formula or theorem.

ODEs are central in physics because they express how a measurable quantity changes with one input, usually time or one spatial coordinate.

Notation

\(x\)

independent variable or variables for this topic

\(y(x)\)

main dependent quantity, field, or function being studied

\(parameter\)

constant that sets a scale, rate, coefficient, or boundary value

\(domain\)

set of input values where the formula or model is used

Method

Step 1: State the object being studied

Name the function, field, signal, or region. State its domain and the units of the physical quantities before doing any algebra or calculus.

Step 2: Apply the central relation

Use the defining relation for Separable Equations:

Separable equation

\[\frac{dy}{dx}=f(x)g(y)\]

Name the task

\[Separable Equations\]

Use the central relation

\[\frac{dy}{dx}=f(x)g(y)\]

Interpret the result

\[Separable equation\]

Step 3: Interpret the result

Translate the mathematical output back into the physical setting. Check whether it represents a rate, amplitude, density, source strength, boundary value, or approximation.

Rules

Separable equation

\[\frac{dy}{dx}=f(x)g(y)\]

Domain reminder

\[\text{formula applies on the stated domain}\]

Units reminder

\[\text{units must balance on both sides}\]

Examples

Question

Identify the central relation for Separable Equations.

Answer

The central relation is Separable equation: \frac{dy}{dx}=f(x)g(y). Use it after naming the variables and checking the assumptions.

Checks

Record equilibrium solutions before dividing by a factor containing the dependent variable.
Define every variable before substituting numbers or interpreting a graph.
Check units, domain restrictions, and sign conventions before trusting the result.

Boundary Conditions

Homogeneous Equations