General solution of differential equation `x^(2)(x+y(dy)/(dx))+(x(dy)/(dx)-y)sqrt(x^(2)+y^(2))=0` is

Options

A. `(1)/(sqrt(x^(2)+y^(2)))+(y)/(x)=c`
B. `sqrt(x^(2)+y^(2))-(y)/(x) = c`
C. `sqrt(x^(2)+y^(2))+(y)/(x)=c`
D. `2 sqrt(x^(2)+y^(2))+(y)/(x)=c`

Explanation

To solve the differential equation

\[ x^{2}\left(x + y \frac{dy}{dx}\right) + \left(x \frac{dy}{dx} - y\right) \sqrt{x^{2} + y^{2}} = 0, \]

we will follow these steps:

### Step 1: Expand the Equation
First, we expand the given equation:

\[ x^{3} + x^{2}y \frac{dy}{dx} + x \frac{dy}{dx} \sqrt{x^{2} + y^{2}} - y \sqrt{x^{2} + y^{2}} = 0. \]

### Step 2: Multiply by \(dx\)
Next, we multiply the entire equation by \(dx\):

\[ x^{3} dx + x^{2}y dy + x \sqrt{x^{2} + y^{2}} dy - y \sqrt{x^{2} + y^{2}} dx = 0. \]

### Step 3: Rearrange the Terms
Rearranging the terms, we can group them:

\[ x^{3} dx + \left(x^{2}y + x \sqrt{x^{2} + y^{2}}\right) dy - y \sqrt{x^{2} + y^{2}} dx = 0. \]

### Step 4: Factor Out Common Terms
We can factor out common terms:

\[ x^{2} \left(x dx + y dy\right) + \sqrt{x^{2} + y^{2}} \left(x dy - y dx\right) = 0. \]

### Step 5: Divide by \(x^{2} \sqrt{x^{2} + y^{2}}\)
Now, we divide the entire equation by \(x^{2} \sqrt{x^{2} + y^{2}}\):

\[ \frac{x dx + y dy}{\sqrt{x^{2} + y^{2}}} + \frac{x dy - y dx}{x^{2}} = 0. \]

### Step 6: Rewrite in Differential Form
This can be rewritten as:

\[ \frac{1}{2} d(x^{2} + y^{2}) + d\left(\frac{y}{x}\right) = 0. \]

### Step 7: Integrate Both Sides
Integrating both sides gives us:

\[ \frac{1}{2} \sqrt{x^{2} + y^{2}} + \frac{y}{x} = C, \]

where \(C\) is a constant of integration.

### Step 8: Final Rearrangement
Rearranging gives us the general solution of the differential equation:

\[ \sqrt{x^{2} + y^{2}} + \frac{2y}{x} = C. \]

### Summary of Steps
1. Expand the equation.
2. Multiply by \(dx\).
3. Rearrange the terms.
4. Factor out common terms.
5. Divide by \(x^{2} \sqrt{x^{2} + y^{2}}\).
6. Rewrite in differential form.
7. Integrate both sides.
8. Rearrange to find the general solution.