Answer :
Answer:
To solve the differential equation \( x \, dx + y \, dy = X \, dY - y \, dx \) by dividing both sides by \( X^2 + y^2 \), let's proceed step by step:
Given differential equation:
\[ x \, dx + y \, dy = X \, dY - y \, dx \]
First, rearrange terms to group similar variables:
\[ x \, dx + y \, dy + y \, dx = X \, dY \]
Combine like terms:
\[ x \, dx + (y + y) \, dx = X \, dY \]
\[ x \, dx + 2y \, dx = X \, dY \]
\[ (x + 2y) \, dx = X \, dY \]
Now, divide both sides by \( X^2 + y^2 \):
\[ \frac{x + 2y}{X^2 + y^2} \, dx = \frac{X}{X^2 + y^2} \, dY \]
Integrate both sides:
Left-hand side:
\[ \int \frac{x + 2y}{X^2 + y^2} \, dx = \frac{1}{X^2 + y^2} \left( x \int dx + 2y \int dx \right) \]
\[ = \frac{1}{X^2 + y^2} \left( x^2 + 2xy \right) \]
Right-hand side:
\[ \int \frac{X}{X^2 + y^2} \, dY = \frac{1}{X^2 + y^2} \int X \, dY \]
\[ = \frac{1}{X^2 + y^2} \left( XY - \int Y \, dX \right) \]
Therefore, the solution to the differential equation \( x \, dx + y \, dy = X \, dY - y \, dx \) divided by \( X^2 + y^2 \) is:
\[ \frac{x^2 + 2xy}{X^2 + y^2} = \frac{XY - \int Y \, dX}{X^2 + y^2} + C \]
where \( C \) is the constant of integration.