Solution of the differential equation `x dy -y dx = sqrt("")(x^(2)+y^(2))dx` is

Options

A. `[y+sqrt("")(x^(2)+y^(2))]=ky^(2)`
B. `[y+sqrt("")(x^(2)+y^(2))]=kx^(2)`
C. `[y+sqrt("")(x^(2)+y^(2))]=k(x^(2)+y^(2))`
D. None of these

Answer

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

### Step 1: Rearranging the Equation
We start by rearranging the given equation to isolate \( dy \) and \( dx \):
\[ x \, dy = y \, dx + \sqrt{x^2 + y^2} \, dx \]
This simplifies to:
\[ x \, dy = (y + \sqrt{x^2 + y^2}) \, dx \]
Now, we can express \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{y + \sqrt{x^2 + y^2}}{x} \]

### Step 2: Homogeneous Function
The equation is a homogeneous function. To solve it, we will use the substitution \( y = vx \), where \( v \) is a function of \( x \). Thus, we have:
\[ dy = v \, dx + x \, dv \]
Substituting \( y = vx \) into the equation gives:
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]

### Step 3: Substituting into the Equation
Substituting \( y = vx \) into the rearranged equation:
\[ v + x \frac{dv}{dx} = \frac{vx + \sqrt{x^2 + (vx)^2}}{x} \]
This simplifies to:
\[ v + x \frac{dv}{dx} = v + \sqrt{1 + v^2} \]
Cancelling \( v \) from both sides:
\[ x \frac{dv}{dx} = \sqrt{1 + v^2} \]

### Step 4: Separating Variables
Now we separate the variables:
\[ \frac{dv}{\sqrt{1 + v^2}} = \frac{dx}{x} \]

### Step 5: Integrating Both Sides
Integrating both sides:
\[ \int \frac{dv}{\sqrt{1 + v^2}} = \int \frac{dx}{x} \]
The left side integrates to \( \ln |v + \sqrt{1 + v^2}| \) and the right side integrates to \( \ln |x| + C \):
\[ \ln |v + \sqrt{1 + v^2}| = \ln |x| + C \]

### Step 6: Exponentiating Both Sides
Exponentiating both sides gives:
\[ |v + \sqrt{1 + v^2}| = k |x| \]
where \( k = e^C \).

### Step 7: Substituting Back for \( v \)
Recall that \( v = \frac{y}{x} \):
\[ \left|\frac{y}{x} + \sqrt{1 + \left(\frac{y}{x}\right)^2}\right| = k |x| \]
This simplifies to:
\[ |y + \sqrt{x^2 + y^2}| = k x^2 \]

### Final Result
Thus, the solution to the differential equation is:
\[ y + \sqrt{x^2 + y^2} = k x^2 \]