Solution of the differential equation x dy -y dx = 0 represents-

Options

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

Explanation

To solve the differential equation \( x \, dy - y \, dx = 0 \), we can rearrange it into a more standard form.

### Step 1: Rearranging the Equation
We start with the equation:
\[ x \, dy - y \, dx = 0 \]
Rearranging gives:
\[ x \, dy = y \, dx \]

### Step 2: Dividing by \( xy \)
Next, we divide both sides by \( xy \) (assuming \( x \neq 0 \) and \( y \neq 0 \)):
\[ \frac{dy}{y} = \frac{dx}{x} \]

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

### Step 4: Exponentiating Both Sides
To eliminate the logarithm, we exponentiate both sides:
\[ |y| = e^{\ln |x| + C} = |x| e^C \]
Let \( k = e^C \) (a positive constant), we can write:
\[ |y| = k |x| \]

### Step 5: Removing Absolute Values
This can be simplified to:
\[ y = kx \quad \text{or} \quad y = -kx \]
Thus, the general solution of the differential equation is:
\[ y = Cx \]
where \( C \) can be any real number (positive or negative).

### Conclusion
The solution \( y = Cx \) represents a family of straight lines passing through the origin.

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