Suppose that we have the following non-lienar homogeneous differential equation, that is
$$ay^{'}(x)+by(x)y^{'}(x)-y(x)=0$$ where $a,b\in\mathbb{R}-{0}$ constants. Considering that $y^{'}\neq0$ the d.e. is written equilevantely as follows
$$a\frac{y^{'}(x)}{y(x)}+by^{'}(x)-1=0\Rightarrow a\ln|y(x)|+by(x)-x=C$$ where $C$ is a constant. Supposing that $y(x)<0$ (or positive) is there any transformation that can give an exact solution $y(x)$? We further asssume that $y(\cdot)$ is a decreasing and continuous differentiable function of $x$.