Solution of First Order and First Degree Differential Equations

Having seen how a differential equation is formed and what it means to solve one, the chapter now develops concrete methods for actually solving first-order, first-degree differential equations — equations of the general shape $\dfrac{dy}{dx}=f(x,y)$ where the derivative appears only to the first power. Three standard techniques are covered, in increasing order of generality: the variables separable method (§10.6.1, when $x$ and $y$ can be algebraically split apart), the substitution method (§10.6.2, for equations of the form $f(ax+by+c)$, reduced to variables separable by a linear substitution), and the homogeneous-equation method (§10.6.3, for equations expressible as a function of $\dfrac{y}{x}$ alone, reduced to variables separable by the substitution $y=vx$ …