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.

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