Find the differential Equation of xydy+ ydx

The given expression is:

\[
x y \, dy + y \, dx
\]

We aim to find a differential equation from this expression. To simplify the expression, divide both sides by \( y \):

\[
x \, dy + dx = 0
\]

This is a first-order differential equation. Now, we can rearrange it as:

\[
\frac{dy}{dx} = -\frac{1}{x}
\]

The general solution to this equation is obtained by integrating both sides:

\[
\int dy = \int -\frac{1}{x} \, dx
\]

This gives:

\[
y = -\ln |x| + C
\]

Where \( C \) is the constant of integration.

Thus, the differential equation is:

\[
x \, dy + y \, dx = 0
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. How would you solve a differential equation like this with an initial condition?
2. Can you explain how the constant of integration affects the solution?
3. What happens if \( x = 0 \) in this differential equation?
4. How do you interpret the solution \( y = -\ln |x| + C \) geometrically?
5. What is the difference between solving homogeneous and non-homogeneous differential equations?

**Tip:** Always check whether a differential equation can be separated into variables for easier integration.

Find the differential equation of xydy + ydx.

Explore the solution of the differential equation x y dy + y dx = 0. Learn how to separate variables, integrate both sides, and derive the general solution. Suitable for advanced high school and early university students.

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