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

Options

A. `x+sqrt(x^(2)+y^(2))=Cy^(2)`
B. `y+sqrt(x^(2)+y^(2))=Cy^(2)`
C. `x+sqrt(x^(2)+y^(2))=Cx^(2)`
D. `y+sqrt(x^(2)+y^(2))=Cx^(2)`

Explanation

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

### Step 1: Rewrite the Equation
Start with the given equation:
\[ x \frac{dy}{dx} = y + \sqrt{x^2 + y^2} \]
Now, divide both sides by \( x \):
\[ \frac{dy}{dx} = \frac{y}{x} + \frac{\sqrt{x^2 + y^2}}{x} \]

### Step 2: Substitute \( y = vx \)
Let \( y = vx \), where \( v \) is a function of \( x \). Then, differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]
Substituting this into the equation gives:
\[ v + x \frac{dv}{dx} = \frac{vx}{x} + \frac{\sqrt{x^2 + (vx)^2}}{x} \]
This simplifies to:
\[ v + x \frac{dv}{dx} = v + \sqrt{1 + v^2} \]

### Step 3: Simplify the Equation
Now, we can cancel \( v \) from both sides:
\[ x \frac{dv}{dx} = \sqrt{1 + v^2} \]

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

### Step 5: Integrate Both Sides
Integrate both sides:
\[ \int \frac{dv}{\sqrt{1 + v^2}} = \int \frac{dx}{x} \]
The left side integrates to:
\[ \ln(v + \sqrt{1 + v^2}) + C_1 \]
The right side integrates to:
\[ \ln|x| + C_2 \]
Setting \( C = C_2 - C_1 \), we have:
\[ \ln(v + \sqrt{1 + v^2}) = \ln|x| + C \]

### Step 6: Exponentiate Both Sides
Exponentiating gives:
\[ v + \sqrt{1 + v^2} = Kx \]
where \( K = e^C \).

### Step 7: Substitute Back for \( v \)
Recall that \( v = \frac{y}{x} \):
\[ \frac{y}{x} + \sqrt{1 + \left(\frac{y}{x}\right)^2} = Kx \]
Multiplying through by \( x \):
\[ y + \sqrt{y^2 + x^2} = Kx^2 \]

### Step 8: Final Rearrangement
This can be rearranged to:
\[ y + \sqrt{x^2 + y^2} = Cx^2 \]
where \( C \) is a constant.

### Conclusion
The solution to the differential equation is:
\[ y + \sqrt{x^2 + y^2} = Cx^2 \]