STATEMENT -1 : The differential equation `(dy)/(dx) = (2xy)/(x^(2) + y^(2))` Can't be solved by the substitution x = vy.
and
STATEMENT-2 : When the differential equation is homogeneous of first order and first degree, then the substitution that solves the equation is y = vx.

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is true, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the given differential equation and analyze the statements, let's proceed step by step.

### Step 1: Identify the Differential Equation
The given differential equation is:

\[ \frac{dy}{dx} = \frac{2xy}{x^2 + y^2} \]

### Step 2: Check if the Equation is Homogeneous
A differential equation is homogeneous if it can be expressed in the form \( \frac{dy}{dx} = f\left(\frac{y}{x}\right) \).

To check if it's homogeneous, we can rewrite the right-hand side:

\[ \frac{2xy}{x^2 + y^2} = \frac{2 \cdot \frac{y}{x} \cdot x^2}{x^2 + \left(\frac{y}{x} \cdot x\right)^2} = \frac{2v}{1 + v^2} \]

where \( v = \frac{y}{x} \).

Since we can express the equation in terms of \( v \), it confirms that the equation is homogeneous.

### Step 3: Use the Substitution \( y = vx \)
To solve the equation, we use the substitution \( y = vx \), where \( v \) is a function of \( x \). Then, we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = v + x\frac{dv}{dx} \]

### Step 4: Substitute into the Differential Equation
Now, substitute \( y = vx \) and \( \frac{dy}{dx} = v + x\frac{dv}{dx} \) into the original equation:

\[ v + x\frac{dv}{dx} = \frac{2x(vx)}{x^2 + (vx)^2} \]

This simplifies to:

\[ v + x\frac{dv}{dx} = \frac{2vx^2}{x^2(1 + v^2)} = \frac{2v}{1 + v^2} \]

### Step 5: Rearranging the Equation
Rearranging gives us:

\[ x\frac{dv}{dx} = \frac{2v}{1 + v^2} - v \]

This can be further simplified to:

\[ x\frac{dv}{dx} = \frac{2v - v(1 + v^2)}{1 + v^2} = \frac{v(1 - v^2)}{1 + v^2} \]

### Step 6: Separate Variables
Now, we can separate variables:

\[ \frac{1 + v^2}{v(1 - v^2)} dv = \frac{1}{x} dx \]

### Step 7: Integrate Both Sides
Integrate both sides to find \( v \):

\[ \int \frac{1 + v^2}{v(1 - v^2)} dv = \int \frac{1}{x} dx \]

### Step 8: Solve the Integrals
The left-hand side can be solved using partial fractions, and the right-hand side integrates to \( \ln|x| + C \).

### Conclusion
The substitution \( y = vx \) is valid for solving this homogeneous differential equation, contradicting Statement 1. Thus, Statement 1 is false, and Statement 2 is true.