Solution of the differential equation
`(sqrtxdx+sqrtydy)/(sqrtxdx-sqrtydy)=sqrt((y^(3))/(x^(3)))` is given by

Options

A. `(3)/(2)log((y)/(x))+log|(x^(3//2)+y^(3//2))/(x^(3//2))|+tan^(-1)((y)/(x))^(3//2)+C=0`
B. `(2)/(3)log((y)/(x))+log|(x^(3//2)+y^(3//2))/(x^(3//2))|+tan^(-1).(y)/(x)+C=0`
C. `(2)/(3)log((y)/(x))+log((x+y)/(x))+tan^(-1)((y^(3//2))/(x^(3//2)))+C=0`
D. none of these

Explanation

To solve the differential equation

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

we will follow these steps:

### Step 1: Rewrite the equation

First, we can rewrite the equation in a more manageable form. We can multiply both sides by the denominator:

\[ \sqrt{x} \, dx + \sqrt{y} \, dy = \sqrt{\frac{y^3}{x^3}} (\sqrt{x} \, dx - \sqrt{y} \, dy). \]

### Step 2: Simplify the right-hand side

Next, we simplify the right-hand side:

\[ \sqrt{x} \, dx + \sqrt{y} \, dy = \frac{y^{3/2}}{x^{3/2}} (\sqrt{x} \, dx - \sqrt{y} \, dy). \]

### Step 3: Rearranging terms

Now, we rearrange the terms to isolate \(dx\) and \(dy\):

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

### Step 4: Factor out common terms

We can factor out common terms from both sides:

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

### Step 5: Introduce substitutions

Let \(u = x^{3/2}\) and \(v = y^{3/2}\). Then, we have:

\[ du = \frac{3}{2} x^{1/2} dx \quad \text{and} \quad dv = \frac{3}{2} y^{1/2} dy. \]

### Step 6: Substitute into the equation

Substituting \(du\) and \(dv\) into the equation gives us:

\[ \frac{du}{u} + \frac{dv}{v} = 0. \]

### Step 7: Integrate both sides

Integrating both sides results in:

\[ \ln |u| + \ln |v| = C, \]

which simplifies to:

\[ \ln |uv| = C. \]

### Step 8: Exponentiate to remove the logarithm

Exponentiating both sides gives:

\[ uv = k, \]

where \(k = e^C\).

### Step 9: Substitute back for \(u\) and \(v\)

Substituting back for \(u\) and \(v\):

\[ x^{3/2} y^{3/2} = k, \]

or

\[ xy = k^2. \]

### Final Result

Thus, the solution of the differential equation is:

\[ x^{3/2} + y^{3/2} = C, \]

where \(C\) is a constant.

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