Solution of the differential equation `xdy+ydx=0` represents a

Options

A. parabola
B. circle
C. hyperbola
D. straight line

Answer

To solve the differential equation \( x \, dy + y \, dx = 0 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x \, dy + y \, dx = 0 \]
We can rearrange this equation to isolate the differentials:
\[ x \, dy = -y \, dx \]

### Step 2: Separating Variables
Next, we will separate the variables \( y \) and \( x \):
\[ \frac{dy}{y} = -\frac{dx}{x} \]

### Step 3: Integrating Both Sides
Now, we will integrate both sides:
\[ \int \frac{dy}{y} = \int -\frac{dx}{x} \]
This gives us:
\[ \log |y| = -\log |x| + C \]
where \( C \) is the constant of integration.

### Step 4: Simplifying the Equation
We can rewrite the right-hand side using properties of logarithms:
\[ \log |y| + \log |x| = C \]
This can be combined into a single logarithm:
\[ \log |xy| = C \]

### Step 5: Exponentiating Both Sides
To eliminate the logarithm, we exponentiate both sides:
\[ |xy| = e^C \]
Let \( k = e^C \) (where \( k \) is a positive constant), we have:
\[ xy = k \]

### Conclusion
The equation \( xy = k \) represents a hyperbola.

### Final Answer
The solution of the differential equation \( x \, dy + y \, dx = 0 \) represents a hyperbola.

---