By elimating the constant in the following equation ` x^2 - y^2 = C ( x^2 + y^2) ^2 ` its differeential equation is ` Y' ( x ( lamda y^2 - x^2 ))/( y ( lamda x^2 - Y ^2)),` then the value of `lamda ` is ….

Answer

To solve the problem step by step, we will start from the given equation and differentiate it to find the value of \( \lambda \).

### Step 1: Start with the given equation
The equation provided is:
\[ x^2 - y^2 = C (x^2 + y^2)^2 \]

### Step 2: Differentiate both sides with respect to \( x \)
We will differentiate the left-hand side and the right-hand side of the equation.

Differentiating the left-hand side:
\[ \frac{d}{dx}(x^2 - y^2) = 2x - 2y \frac{dy}{dx} = 2x - 2y y' \]

Differentiating the right-hand side using the product and chain rule:
\[ \frac{d}{dx}(C(x^2 + y^2)^2) = C \cdot 2(x^2 + y^2) \cdot \frac{d}{dx}(x^2 + y^2) = C \cdot 2(x^2 + y^2)(2x + 2y y') \]

### Step 3: Set the derivatives equal to each other
Equating the derivatives from both sides, we have:
\[ 2x - 2y y' = 2C(x^2 + y^2)(2x + 2y y') \]

### Step 4: Simplify the equation
We can simplify the equation by dividing by 2:
\[ x - y y' = C(x^2 + y^2)(2x + 2y y') \]

### Step 5: Rearranging the equation
Rearranging gives us:
\[ x - y y' = 2C(x^2 + y^2)(x + y y') \]

### Step 6: Isolate \( y' \)
To isolate \( y' \), we can rearrange the equation:
\[ x - 2C(x^2 + y^2)x = y y' + 2C(x^2 + y^2)y y' \]
\[ x - 2C(x^2 + y^2)x = y y'(1 + 2C(x^2 + y^2)) \]
\[ y' = \frac{x - 2C(x^2 + y^2)x}{y(1 + 2C(x^2 + y^2))} \]

### Step 7: Compare with the given form of \( y' \)
We are given that:
\[ y' = \frac{x(\lambda y^2 - x^2)}{y(\lambda x^2 - y^2)} \]

### Step 8: Set the two expressions for \( y' \) equal
Now we can compare the two expressions for \( y' \):
\[ \frac{x - 2C(x^2 + y^2)x}{y(1 + 2C(x^2 + y^2))} = \frac{x(\lambda y^2 - x^2)}{y(\lambda x^2 - y^2)} \]

### Step 9: Eliminate \( y \) and simplify
By cross-multiplying and simplifying, we can find \( \lambda \):
\[ (x - 2C(x^2 + y^2)x)(\lambda x^2 - y^2) = x(\lambda y^2 - x^2)(1 + 2C(x^2 + y^2)) \]

### Step 10: Solve for \( \lambda \)
After simplifying, we can find that:
\[ \lambda = 3 \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \lambda = 3 \]