Homogeneous Differential Equations

For \(\dfrac{\mathrm dy}{\mathrm dx}=\mathrm{f}\left(\dfrac yx\right)\), use \(y=vx\), where \(v\) is a function of \(x\) and \(x\ne0\).

\(\dfrac{\mathrm dy}{\mathrm dx}=v+x\dfrac{\mathrm dv}{\mathrm dx},\qquad x\dfrac{\mathrm dv}{\mathrm dx}=\mathrm{f}(v)-v\).

Separate the variables. Check constant values of \(v\) satisfying \(\mathrm{f}(v)=v\) before dividing; these give straight-line solutions. Substitute \(v=\dfrac yx\) back into the result.

Example

For \(\dfrac{\mathrm dy}{\mathrm dx}=1+\dfrac yx\), substitution gives \(x\dfrac{\mathrm dv}{\mathrm dx}=1\).

\(v=\ln|x|+C,\qquad y=x(\ln|x|+C)\), on an interval not crossing zero.