First-Order DEs (Separable & Homogeneous)

First-Order DEs (Separable & Homogeneous)

$Learning Objectives$

Classify differential equations by type, order, and linearity.
Understand Initial Value Problems (IVPs) and geometric solutions via slope fields.
Solve first-order differential equations using the separation of variables method.
Identify and solve homogeneous first-order differential equations.

A Differential Equation (DE) is an equation containing an unknown function and one or more of its derivatives. Differential equations are the language of nature, describing how quantities change relative to one another.

$Differential Equation$

An equation involving an independent variable $x$ , a dependent variable $y$ , and one or more derivatives of $y$ with respect to $x$ .

Classifying Differential Equations

Before solving a DE, it is crucial to classify it to determine the appropriate solution method. The first major distinction is between Ordinary and Partial Differential Equations.

$ODE vs. PDE$

Ordinary Differential Equation (ODE): Involves derivatives with respect to only one independent variable. Example: $\frac{dy}{dx} + 5y = e^x$
Partial Differential Equation (PDE): Involves partial derivatives with respect to two or more independent variables. Example: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

$Classification Criteria for ODEs$

Order: The order of a DE is the order of the highest derivative present in the equation.
Degree: The degree of a DE is the power of the highest derivative, provided the equation is a polynomial in its derivatives.
Linearity: A DE is linear if the dependent variable $y$ and all its derivatives appear to the first power and are not multiplied together or acted upon by non-linear functions (like sine, exp).

Examples:

$\frac{dy}{dx} + 5y = e^x$ (First-order, First-degree, Linear)
$\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + y = 0$ (Second-order, First-degree, Non-linear)

Initial Value Problems (IVPs)

A differential equation typically has infinitely many solutions, forming a general solution with arbitrary constants (like $+ C$ ). To find a particular solution, we need additional information.

$Initial Value Problem (IVP)$

An IVP consists of a differential equation coupled with one or more initial conditions evaluated at the same point.

For a first-order DE: $\frac{dy}{dx} = f(x,y)$ subject to $y(x_0) = y_0$ .

Visualizing Solutions: Direction Fields

For a first-order DE $y' = f(x,y)$ , we can visualize the solution without solving it analytically by plotting the slope $f(x,y)$ at various points $(x,y)$ . This creates a Slope Field (or Direction Field).

$Interactive Simulation$

Interact with the slope field below to visualize solutions for separable and homogeneous equations.

Equation Type

Initial Condition $(x_0, y_0)$

x_0

1.0

y_0

1.0

Drag the sliders to change the initial point $(x_0, y_0)$ and see how the particular solution follows the direction field.

Variable Separable Differential Equations

The simplest type of first-order DE is one where the variables can be separated on opposite sides of the equation.

$Separable Equation$

A first-order DE is separable if it can be factored into the form:

N(y) \, dy = M(x) \, dx

$Method of Separation of Variables$

STEP-BY-STEP

Separate: Rearrange terms so that all $y$ 's are with $dy$ and all $x$ 's are with $dx$ . If division removes values of $y$ , test those excluded constant solutions in the original equation.
Integrate: Integrate both sides: $\int N(y) \, dy = \int M(x) \, dx$ .
Solve: Add an arbitrary constant $C$ to one side and solve for $y$ explicitly if possible. If an IVP is given, plug in the initial conditions to solve for $C$ .

Homogeneous Differential Equations

A first-order DE is homogeneous if it can be written in the form $M(x,y)dx + N(x,y)dy = 0$ where $M$ and $N$ are homogeneous functions of the same degree.

$Homogeneous Function Test$

A function $f(x,y)$ is homogeneous of degree $n$ if for any parameter $\lambda$ :

f(\lambda x, \lambda y) = \lambda^n f(x,y)

Tip: If all terms in $M$ and $N$ have the same total degree of variables, the DE is homogeneous.

$Solving Homogeneous DEs$

STEP-BY-STEP

Substitute: Let $y = vx$ $y = v x$ (implies $dy = vdx + xdv$ $d y = v d x + x d v$ ) OR $x = vy$ $x = v y$ (implies $dx = vdy + ydv$ $d x = v d y + y d v$ ).
- Use $y = vx$ if $N$ is simpler.
- Use $x = vy$ if $M$ is simpler.
Simplify: Substitute these into the original DE. The new equation in terms of $v$ and $x$ (or $y$ ) will be separable.
Integrate: Separate variables and integrate.
Back-Substitute: Replace $v$ with $y/x$ (or $x/y$ ) to get the final general solution.

Visual Relationships in First-Order Equations

A Consistent Order for Classifying an ODE

Classify the derivative structure before choosing a solution technique; degree is defined only when the equation is polynomial in its derivatives.

Identify one independent variable: ordinary derivatives → ODE

Record the highest derivative order

For a polynomial equation, record the highest-derivative degree

Check whether the dependent variable and derivatives enter linearly

An Initial Condition Selects a Solution Curve

A first-order general solution is a family of curves; an initial value selects the member that passes through the specified point when a unique solution exists.

Solve the DE to obtain a one-constant family

Locate the prescribed point (x₀, y₀)

Choose the family member satisfying y(x₀) = y₀

Substitute the selected solution into both the DE and the condition

Separation Requires an Excluded-Solution Check

Dividing by a function of y can remove equilibrium solutions, so test excluded zero factors in the original differential equation.

Write y′ = g(x)h(y)

On intervals where h(y) ≠ 0, separate dy/h(y) = g(x)dx

Integrate on the interval and apply any initial value

Check each constant solution h(y) = 0 in the original equation

Homogeneous Substitution Uses a Variable Ratio

A homogeneous first-order equation depends on a ratio of variables; either ratio substitution reduces it to a separable equation when used on a valid interval.

Check that M and N are homogeneous of the same degree

Choose y = vx or x = vy to simplify the equation

Differentiate the substitution and reduce to separable form

Integrate and back-substitute on an interval where the ratio is defined

Key Takeaways

A Differential Equation relates a function to its derivatives. ODEs have one independent variable; PDEs have multiple.
An Initial Value Problem (IVP) provides conditions to find a specific particular solution from the general solution.
Separation of Variables is the primary method for solving first-order DEs where $x$ and $y$ terms can be isolated.
Homogeneous DEs can be transformed into separable equations using the substitution $y=vx$ or $x=vy$ .
Slope Fields provide a geometric interpretation of solutions without solving analytically.
Always check the order and degree to classify the equation correctly before starting.