Solution of the differential equation `(xdy)/(x^(2)+y^(2))=((y)/(x^(2)+y^(2))-1)dx`, is

Answer

To solve the differential equation

\[ \frac{x \, dy}{x^2 + y^2} = \left(\frac{y}{x^2 + y^2} - 1\right) dx, \]

we will follow these steps:

### Step 1: Rearranging the Equation
First, we can multiply both sides by \(x^2 + y^2\) to eliminate the denominator:

\[ x \, dy = \left(y - (x^2 + y^2)\right) dx. \]

This simplifies to:

\[ x \, dy = (y - x^2 - y^2) dx. \]

### Step 2: Rearranging Terms
Next, we can rearrange the equation:

\[ x \, dy - y \, dx = -x^2 \, dx - y^2 \, dx. \]

### Step 3: Factoring
Now, we can factor out the common terms:

\[ x \, dy - y \, dx = -\left(x^2 + y^2\right) dx. \]

### Step 4: Dividing by \(x^2 + y^2\)
Now, we can divide the entire equation by \(x^2 + y^2\):

\[ \frac{x \, dy - y \, dx}{x^2 + y^2} = -dx. \]

### Step 5: Recognizing the Differential Form
Notice that the left side can be recognized as the differential of \(\tan^{-1}\left(\frac{y}{x}\right)\):

\[ d\left(\tan^{-1}\left(\frac{y}{x}\right)\right) = \frac{x \, dy - y \, dx}{x^2 + y^2}. \]

### Step 6: Integrating Both Sides
Now we can integrate both sides:

\[ \int d\left(\tan^{-1}\left(\frac{y}{x}\right)\right) = \int -dx. \]

This gives us:

\[ \tan^{-1}\left(\frac{y}{x}\right) = -x + C, \]

where \(C\) is the constant of integration.

### Step 7: Rearranging the Solution
Finally, we can express the solution in a more standard form:

\[ \tan^{-1}\left(\frac{y}{x}\right) + x = C. \]

### Final Solution
Thus, the general solution of the given differential equation is:

\[ \tan^{-1}\left(\frac{y}{x}\right) + x = C. \]

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