The general solution of `x^2(dy)/(dx)=2 ` is

Options

A. `y=c+2/x`
B. `y=c-2/x`
C. y=2cx
D. `y=c-3/x^2`

Answer

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

### Step 1: Rearranging the equation
We start with the given equation:
\[ x^2 \frac{dy}{dx} = 2 \]
We can rearrange this to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{2}{x^2} \]

### Step 2: Integrating both sides
Next, we will integrate both sides. The left side integrates with respect to \( y \), and the right side integrates with respect to \( x \):
\[ \int dy = \int \frac{2}{x^2} dx \]

### Step 3: Performing the integration
The left side integrates to:
\[ y = \int dy = y + C_1 \]
The right side integrates to:
\[ \int \frac{2}{x^2} dx = 2 \int x^{-2} dx = 2 \left( -\frac{1}{x} \right) = -\frac{2}{x} + C_2 \]

### Step 4: Combining the results
Combining both sides, we have:
\[ y = -\frac{2}{x} + C \]
where \( C = C_2 - C_1 \) is a constant of integration.

### Step 5: Final form of the general solution
Thus, the general solution of the differential equation is:
\[ y = -\frac{2}{x} + C \]