Homogeneous First-Order Differential Equations

A first-order equation of the form \(\frac{\mathrm dy}{\mathrm dx}=F(y/x)\) is homogeneous. For \(x\ne0\), set \(y=vx\), where \(v\) depends on \(x\).

Differentiate the substitution: \(\frac{\mathrm dy}{\mathrm dx}=v+x\frac{\mathrm dv}{\mathrm dx}\). Substitute this into the equation, separate the variables, then integrate.

Use the initial condition to find the constant. Finally check the interval on which the solution satisfies the original equation and any positivity restrictions.