6.2 Integrable first-order differential equations

We begin by considering differential equations of the form:

where $f(x)$ is a function of $x$. A solution to this equation is given by $y=F(x)$, where $F(x)$ is a function with $F^{\prime}(x)=f(x)$. By the fundamental theorem of calculus, we can obtain $F(x)$ by integrating $f(x)$. Differential equations of this form are called integrable.

Solution. We integrate $1/(x-2)$ to get $y={\log|x-2|}+c$ where $c$ is a constant.