The solution of `(dy)/(dx) = (y+sqrt(x^(2) -y^(2)))/x ` is

Options

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

Answer

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

### Step 1: Identify the type of differential equation
The given equation is a homogeneous differential equation because the degrees of the numerator and denominator are the same.

### Step 2: Substitute \(y = vx\)
We will use the substitution \(y = vx\), where \(v\) is a function of \(x\). This gives us:
\[ \frac{dy}{dx} = v + x\frac{dv}{dx} \]

### Step 3: Substitute into the original equation
Now, substituting \(y = vx\) into the original equation:
\[ v + x\frac{dv}{dx} = \frac{vx + \sqrt{x^2 - (vx)^2}}{x} \]
This simplifies to:
\[ v + x\frac{dv}{dx} = v + \sqrt{1 - v^2} \]

### Step 4: Simplify the equation
We can cancel \(v\) from both sides:
\[ x\frac{dv}{dx} = \sqrt{1 - v^2} \]

### Step 5: Separate variables
Now, we separate the variables \(v\) and \(x\):
\[ \frac{dv}{\sqrt{1 - v^2}} = \frac{dx}{x} \]

### Step 6: Integrate both sides
Integrating both sides:
\[ \int \frac{dv}{\sqrt{1 - v^2}} = \int \frac{dx}{x} \]
The left-hand side integrates to \(\sin^{-1}(v)\) and the right-hand side integrates to \(\log|x| + C\):
\[ \sin^{-1}(v) = \log|x| + C \]

### Step 7: Substitute back for \(v\)
Recall that \(v = \frac{y}{x}\), so we substitute back:
\[ \sin^{-1}\left(\frac{y}{x}\right) = \log|x| + C \]

### Final Solution
Thus, the solution to the differential equation is:
\[ \sin^{-1}\left(\frac{y}{x}\right) = \log|x| + C \]