The solution of the differential equation :
`2x (dy)/(dx)-y=3` represents a family of :

Options

A. straight lines
B. circles
C. parabolas
D. ellipses.

Answer

To solve the differential equation \( 2x \frac{dy}{dx} - y = 3 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ 2x \frac{dy}{dx} - y = 3 \]
Rearranging this, we can express \( \frac{dy}{dx} \) in terms of \( y \) and \( x \):
\[ 2x \frac{dy}{dx} = y + 3 \]
\[ \frac{dy}{dx} = \frac{y + 3}{2x} \]

### Step 2: Separating Variables
Next, we separate the variables \( y \) and \( x \):
\[ \frac{dy}{y + 3} = \frac{dx}{2x} \]

### Step 3: Integrating Both Sides
Now we integrate both sides:
\[ \int \frac{dy}{y + 3} = \int \frac{dx}{2x} \]
The left side integrates to:
\[ \ln |y + 3| \]
And the right side integrates to:
\[ \frac{1}{2} \ln |x| + C \]
where \( C \) is the constant of integration.

### Step 4: Combining the Results
Combining the results from the integration, we have:
\[ \ln |y + 3| = \frac{1}{2} \ln |x| + C \]
Exponentiating both sides to eliminate the logarithm gives:
\[ |y + 3| = e^C |x|^{1/2} \]
Let \( k = e^C \), then we can write:
\[ y + 3 = k \sqrt{x} \]

### Step 5: Solving for \( y \)
Finally, we solve for \( y \):
\[ y = k \sqrt{x} - 3 \]

### Step 6: Identifying the Family of Curves
The equation \( y = k \sqrt{x} - 3 \) represents a family of curves where \( k \) is a parameter. This is a family of parabolas that open upwards, shifted down by 3 units.

### Conclusion
Thus, the solution of the differential equation represents a family of parabolas.

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