TITLE: Solving $x \frac{\mathrm dy}{\mathrm dx}=y+1$
QUESTION [5 upvotes]: Following differential equation is given:
$$x \frac{\mathrm dy}{\mathrm dx}=y+1.$$
Separating variables and integrating:
$$\int \frac{1}{y+1} \mathrm dy=\int \frac 1x \mathrm dx$$
$$\ln|y+1|=\ln|x|+c$$
In my textbook the following step is:
$$y+1=Ax \,\,\,\,\, \text{(where} \ A=e^c).$$
My question is why the modulus function can be omitted after exponentiating.

REPLY [2 votes]: Without using logs function
$$x \frac{\mathrm dy}{\mathrm dx}=y+1$$
$$xy'-y=1$$
$$(\frac yx)'=\frac 1 {x^2}$$
Integrate
$$\frac yx=\int \frac {dx} {x^2}$$
$$\implies y=Kx-1$$