The degree of the differential equation satisfying by the curves `sqrt(1+x)-asqrt(1+y)=1,` is ……..

Answer

To find the degree of the differential equation satisfied by the curve given by the equation \( \sqrt{1+x} - a\sqrt{1+y} = 1 \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ \sqrt{1+x} - a\sqrt{1+y} = 1 \]
Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}(\sqrt{1+x}) - a\frac{d}{dx}(\sqrt{1+y}) = 0 \]

### Step 2: Apply the chain rule
Using the chain rule, we find the derivatives:
\[ \frac{1}{2\sqrt{1+x}} - a \cdot \frac{1}{2\sqrt{1+y}} \cdot \frac{dy}{dx} = 0 \]

### Step 3: Rearrange to isolate \( \frac{dy}{dx} \)
Rearranging the equation gives:
\[ a \cdot \frac{1}{2\sqrt{1+y}} \cdot \frac{dy}{dx} = \frac{1}{2\sqrt{1+x}} \]
Multiplying both sides by \( 2\sqrt{1+y} \):
\[ a \frac{dy}{dx} = \frac{\sqrt{1+y}}{\sqrt{1+x}} \]

### Step 4: Express \( \frac{dy}{dx} \)
Now we can express \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{\sqrt{1+y}}{a\sqrt{1+x}} \]

### Step 5: Substitute back into the original equation
Now we substitute back into the original equation:
\[ \sqrt{1+x} - a \cdot \left( \frac{\sqrt{1+y}}{a\sqrt{1+x}} \right) = 1 \]
This simplifies to:
\[ \sqrt{1+x} - \frac{\sqrt{1+y}}{\sqrt{1+x}} = 1 \]

### Step 6: Clear the square root by squaring both sides
To eliminate the square root, we square both sides:
\[ \left( \sqrt{1+x} - \frac{\sqrt{1+y}}{\sqrt{1+x}} \right)^2 = 1^2 \]
Expanding this gives:
\[ (1+x) - 2\sqrt{(1+x)(1+y)} + \frac{(1+y)}{(1+x)} = 1 \]

### Step 7: Rearranging and simplifying
Rearranging the equation leads to:
\[ 1+x - 1 - \frac{(1+y)}{(1+x)} = 2\sqrt{(1+x)(1+y)} \]
This can be simplified further, but we are primarily interested in the degree of the resulting differential equation.

### Step 8: Identify the degree
The highest power of \( \frac{dy}{dx} \) in the equation after squaring is 2. Therefore, the degree of the differential equation is:
\[ \text{Degree} = 2 \]

### Final Answer
The degree of the differential equation satisfying the curves is **2**.

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