Solve the equation `ydx+(x-y^2)dy=0`

Explanation

To solve the differential equation \( y \, dx + (x - y^2) \, dy = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate \( dx \) and \( dy \):
\[ y \, dx = y^2 \, dy - x \, dy \]
This can be rewritten as:
\[ dx = \frac{y^2 - x}{y} \, dy \]

### Step 2: Writing in the Form of a Linear Differential Equation
We can express \( \frac{dx}{dy} \) as:
\[ \frac{dx}{dy} = \frac{y^2 - x}{y} \]
This can be rearranged to:
\[ \frac{dx}{dy} + \frac{x}{y} = y \]
This is now in the standard form of a linear differential equation, \( \frac{dx}{dy} + P(y)x = Q(y) \), where \( P(y) = \frac{1}{y} \) and \( Q(y) = y \).

### Step 3: Finding the Integrating Factor
To solve this linear differential equation, we need to find the integrating factor \( \mu(y) \):
\[ \mu(y) = e^{\int P(y) \, dy} = e^{\int \frac{1}{y} \, dy} = e^{\ln |y|} = |y| = y \quad (\text{assuming } y > 0) \]

### Step 4: Multiplying the Equation by the Integrating Factor
Now we multiply the entire equation by the integrating factor \( y \):
\[ y \frac{dx}{dy} + x = y^2 \]

### Step 5: Recognizing the Left Side as a Derivative
The left-hand side can be recognized as the derivative of a product:
\[ \frac{d}{dy}(xy) = y^2 \]

### Step 6: Integrating Both Sides
Now we integrate both sides with respect to \( y \):
\[ \int \frac{d}{dy}(xy) \, dy = \int y^2 \, dy \]
This gives us:
\[ xy = \frac{y^3}{3} + C \]
where \( C \) is the constant of integration.

### Step 7: Solving for \( x \)
Finally, we can solve for \( x \):
\[ x = \frac{y^2}{3} + \frac{C}{y} \]

### Final Solution
The general solution of the differential equation is:
\[ x = \frac{y^2}{3} + \frac{C}{y} \]

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