Integral curve satisfying ` y'=(x^2 +y^2)/(x^2-y^2)` and ` y' (1) ne 1` has the slope at the point (1, 0) of the curve equal to:

Options

A. `-(5)/(3)`
B. `-1`
C. `1`
D. `5/3`

Answer

To find the slope of the integral curve at the point (1, 0) given the differential equation \( y' = \frac{x^2 + y^2}{x^2 - y^2} \), we will follow these steps:

### Step 1: Understand the given differential equation
The equation given is:
\[ y' = \frac{x^2 + y^2}{x^2 - y^2} \]
Here, \( y' \) represents the derivative \( \frac{dy}{dx} \).

### Step 2: Substitute the point (1, 0)
We need to find the slope at the point (1, 0). Thus, we will substitute \( x = 1 \) and \( y = 0 \) into the equation.

### Step 3: Calculate \( y' \) at the point (1, 0)
Substituting \( x = 1 \) and \( y = 0 \) into the equation:
\[ y' = \frac{1^2 + 0^2}{1^2 - 0^2} = \frac{1 + 0}{1 - 0} = \frac{1}{1} = 1 \]

### Step 4: Conclusion
The slope of the curve at the point (1, 0) is:
\[ \text{slope} = 1 \]

Thus, the slope at the point (1, 0) of the curve is equal to **1**.

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