미분방정식 전체 문제

(a) Since $|y| \geq 0$ no matter what $y$ is, we see $Cx \ge 0$ for all $x$ in the domain of $f(x)$. Since $C$ is positive, that means the domain of $f(x)$ only includes nonnegative numbers. So, the largest possible domain of $f(x)$ is $[0,\infty)$. (b) None exists.

The graph of $Cx$ is given below for some positive constant $C$, also with the graph of $-Cx$. If $y=f(x)$ were sometimes the top function, and other times the bottom function, then there would be a jump discontinuity where it switched. Then the derivative of $f(x)$ would not exist, violating the second property.

A tiny technical note is that it's possible that $f(x)=Cx$ when $x=0$ and $f(x)=-Cx$ when $x>0$ (or vice-versa). This would not introduce a jump discontinuity, but it also does not satisfy that $f(x)>0$ for some values of $x$.

Remark: in several instances below, solving a differential equation will lead us to conclude something like $|y|=g(x)$. In these cases, we choose either $y=g(x)$, or $y=-g(x)$, but not $y=\pm g(x)$ (which is not a function) or that $y$ is sometimes $g(x)$, and other times $-g(x)$. The reasoning above somewhat explains this choice: if $y$ were sometimes positive and sometimes negative, then $\frac{\mathrm{d}y}{\mathrm{d}x}$ would not exist at the values of $x$ where the sign of $y$ switches, unless that switch occurrs at a root of $g(x)$. Since that's a pretty specific occurrence, we usually feel safe ignoring it to avoid getting bogged down in technical details.