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 \( xdy - ydx = 0 \) and determine what it represents, we can follow these steps:

### Step 1: Rewrite the equation
We start with the given differential equation:
\[ xdy - ydx = 0 \]
We can rearrange this to:
\[ xdy = ydx \]

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

### Step 3: Integrate both sides
Now, we integrate both sides:
\[ \int \frac{dy}{y} = \int \frac{dx}{x} \]
The integrals yield:
\[ \log |y| = \log |x| + C \]
where \( C \) is the constant of integration.

### Step 4: Simplify the equation
Using the properties of logarithms, we can rewrite the equation:
\[ \log |y| = \log |x| + \log |c| \]
This can be combined into:
\[ \log |y| = \log |cx| \]
where \( c = e^C \) is a positive constant.

### Step 5: Exponentiate both sides
Exponentiating both sides to eliminate the logarithm gives:
\[ |y| = |cx| \]
This implies:
\[ y = cx \]
where \( c \) can be any real number (positive or negative).

### Step 6: Conclusion
The equation \( y = cx \) represents a family of straight lines passing through the origin.

### Final Answer
The solution of the differential equation \( xdy - ydx = 0 \) represents a family of straight lines passing through the origin.

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