Homogeneous differential equations

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School:Mathematics > Topic:Differential_Equations > Ordinary Differential Equations > Homogeneous Differential Equations

Homogeneous

Definition

The word “homogeneous” can mean different things depending on what kind of differential equation you’re working with. A homogeneous equation in this sense is defined as one where the following relationship is true:

$\textstyle f(tx,ty) = t \cdot f(x,y)$

Solution

The solution to a homogeneous equation is to:

Use the substitution $\textstyle y = ux$ where $u$ is a substitution variable.
Implicitly differentiate the above equation to get $\frac {dy} {dx} = x \frac {du} {dx} + u$ .
Replace $\textstyle \frac {dy} {dx}$ and $\textstyle y$ with these expressions.
Solve for $u$ .
Substitute with the expression $u = \frac {y} {x}$ Then solve for $\textstyle y$ .

The advantage of this method is that the function is in terms of 2 variables, but we simplify the equation by relating $\textstyle y$ and $\textstyle x$ to each other.

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