Homogeneous Differential Equations

A Differential Equation is an equation with a function and one or more of its derivatives:

Homogeneous Differential Equations

Example:

an equation with the function y and its derivative Homogeneous Differential Equations Derivative

Homogeneous Differential Equations

A first order Differential Equation is Homogeneous when it can be in this form:

Homogeneous Differential Equations Formula

We can solve it using Separation of Variables but first we create a new variable Homogeneous Differential Equations Formula 2

Homogeneous Differential Equations Formula 3

With Homogeneous Differential Equations Formula 4 we can solve the Differential Equation.

Example
Homogeneous Differential Equations Formula 5

First, get it in Homogeneous Differential Equations Formula 6 form

Homogeneous Differential Equations Formula 7

Now, we have a function of (y/x).

Then, substitute Homogeneous Differential Equations Formula 4

We’ll get

Homogeneous Differential Equations Formula 8

Now use Separation of Variables:

Homogeneous Differential Equations Formula 9

Put C in ln form with C = ln k

Homogeneous Differential Equations Formula 10

Now substitute back Homogeneous Differential Equations Formula 2

Homogeneous Differential Equations Solutions

We got the solution.

Learn More

Implicit Differentiation

Differentiable

Common Derivatives

Integration by Substitution (Reverse Chain Rule)

Calculus Index

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