The solution of differential equation xdy-ydx=0 represents

Options

A. a reactangular hyperbola
B. parabola whose vertex is at orgin
C. straight line passing through origin
D. a circle whose centre is at origin

Explanation

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

### Step 1: Rewrite the Differential Equation
We start with the given equation:
\[ x \, dy - y \, dx = 0 \]
We can rearrange this to isolate \( dy \) and \( dx \):
\[ x \, dy = y \, dx \]

### Step 2: Separate Variables
Next, we can separate the variables by dividing both sides by \( y \) and \( x \):
\[ \frac{dy}{y} = \frac{dx}{x} \]

### Step 3: Integrate 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: Exponentiate to Solve for y
To eliminate the logarithm, we exponentiate both sides:
\[ |y| = e^{\log |x| + C} = |x| e^C \]
Let \( k = e^C \), which is a positive constant, so we can rewrite it as:
\[ y = kx \]

### Step 5: Interpret the Solution
The equation \( y = kx \) represents a family of straight lines passing through the origin. The constant \( k \) can take any real value, which means we have a straight line for each value of \( k \).

### Conclusion
Thus, the solution of the differential equation \( x \, dy - y \, dx = 0 \) represents a straight line passing through the origin.