Solve the following differential equations.
` ( dx) /( x+y) = ( dy)/(x-y)`

Answer

To solve the differential equation

\[ \frac{dx}{x+y} = \frac{dy}{x-y}, \]

we start by cross-multiplying to rearrange the equation:

\[ dx \cdot (x - y) = dy \cdot (x + y). \]

Expanding both sides gives us:

\[ x \, dx - y \, dx = x \, dy + y \, dy. \]

Rearranging the terms leads to:

\[ x \, dx - y \, dy - x \, dy - y \, dx = 0. \]

This can be rewritten as:

\[ x \, dx - y \, dy - (x + y) \, dy = 0. \]

Now, we can group the terms:

\[ x \, dx - (y + x) \, dy = 0. \]

Next, we can express this as a total differential:

\[ d(xy) = x \, dx - y \, dy. \]

Now we integrate both sides:

\[ \int d(xy) = \int 0. \]

This gives us:

\[ xy = C, \]

where \(C\) is a constant of integration.

To express this in a more familiar form, we can rearrange it:

\[ xy = C \implies x \cdot y = C. \]

Thus, the solution to the differential equation is:

\[ x \cdot y = C. \]