The solution of differential equation `xdx+ydy=a(x^(2)+y^(2))dy` is

Options

A. `x^(2)+y^(2)=Ce^(ay)`
B. `x^(2)+y^(2)=Ce^(2ay)`
C. `x^(@)+y^(2)=e^(2Cay)`
D. none of these

Explanation

To solve the differential equation \( x \, dx + y \, dy = a(x^2 + y^2) \, dy \), we will follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the given differential equation:
\[ x \, dx + y \, dy = a(x^2 + y^2) \, dy \]
We can rewrite it as:
\[ x \, dx + y \, dy - a(x^2 + y^2) \, dy = 0 \]

### Step 2: Factor Out \( dy \)
Next, we can factor out \( dy \) from the terms that contain it:
\[ x \, dx + (y - a(x^2 + y^2)) \, dy = 0 \]

### Step 3: Multiply by 2
To simplify the integration, multiply the entire equation by 2:
\[ 2x \, dx + 2y \, dy = 2a(x^2 + y^2) \, dy \]

### Step 4: Rearranging Again
Rearranging gives:
\[ 2x \, dx + 2y \, dy - 2a(x^2 + y^2) \, dy = 0 \]
This can be rewritten as:
\[ 2x \, dx + 2y \, dy = 2a(x^2 + y^2) \, dy \]

### Step 5: Divide by \( x^2 + y^2 \)
Now, we can divide both sides by \( x^2 + y^2 \):
\[ \frac{2x \, dx + 2y \, dy}{x^2 + y^2} = 2a \, dy \]

### Step 6: Recognizing the Derivative
Notice that \( 2x \, dx + 2y \, dy \) is the differential of \( x^2 + y^2 \):
\[ d(x^2 + y^2) = 2x \, dx + 2y \, dy \]
Thus, we can rewrite the equation as:
\[ \frac{d(x^2 + y^2)}{x^2 + y^2} = 2a \, dy \]

### Step 7: Integrate Both Sides
Integrating both sides gives:
\[ \int \frac{d(x^2 + y^2)}{x^2 + y^2} = \int 2a \, dy \]
This results in:
\[ \ln(x^2 + y^2) = 2ay + C \]
where \( C \) is the constant of integration.

### Step 8: Exponentiate to Solve for \( x^2 + y^2 \)
Exponentiating both sides, we have:
\[ x^2 + y^2 = e^{2ay + C} = e^{C} e^{2ay} \]
Let \( k = e^{C} \), then:
\[ x^2 + y^2 = k e^{2ay} \]

### Final Step: General Solution
Thus, the general solution of the differential equation is:
\[ x^2 + y^2 = C e^{2ay} \]
where \( C \) is a constant.