Solve a Separable DE (1 + x) dy-y dx= 0.

(1) $\begin{equation*} (1+x)\times \frac{d y}{d x} - y = 0 \end{equation*}$

(2) $\begin{equation*} (1+x)\timesdy = y\timesdx \end{equation*}$

Dividing by $y(1+x)$ , we get:

(3) $\begin{equation*} \frac{1}{y}dy = \frac{1}{1+x} dx \end{equation*}$

By Integrating, we get:

(4) $\begin{equation*} \int \frac{1}{y} dy = \int \frac{1}{1+x} dx \end{equation*}$

The left-hand side can be integrated as:

(5) $\begin{equation*} \ln|y| = \ln|1+x| + C_1 \end{equation*}$

where $C_1$ is the constant of integration.

(6) $\begin{equation*} |y| = e^{\ln|1+x| + C_1} = C_2 (1+x) \end{equation*}$

where $C_2 = e^{C_1}$ is another constant of integration. Since $|y| = \pm y$ , we can write the solution as:

(7) $\begin{equation*} y = C (1+x) \end{equation*}$

where $C = \pm C_2$ is a constant.

where $C$ is a constant.