The general solution of the differential equation `(y^(2)-x^(3)) dx - xydy = 0(x ne 0)` is: (where c is a constant of integration)

Explanation

To solve the differential equation \((y^2 - x^3)dx - xy dy = 0\), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given equation:
\[ (y^2 - x^3)dx - xy dy = 0 \]
We can rearrange this to:
\[ (y^2 - x^3)dx = xy dy \]

### Step 2: Divide by \(dx\)
Next, we divide the entire equation by \(dx\):
\[ y^2 - x^3 = xy \frac{dy}{dx} \]
This can be rearranged to:
\[ \frac{dy}{dx} = \frac{y^2 - x^3}{xy} \]

### Step 3: Separate Variables
We can separate the variables:
\[ \frac{dy}{dx} = \frac{y}{x} - \frac{x^2}{y} \]
This gives us:
\[ \frac{y}{y^2 - x^3} dy = \frac{1}{x} dx \]

### Step 4: Substitute \(y^2 = v\)
Let \(y^2 = v\), then \(2y \frac{dy}{dx} = \frac{dv}{dx}\) or \(\frac{dy}{dx} = \frac{1}{2y} \frac{dv}{dx}\). Substituting this into our equation, we have:
\[ \frac{1}{2} \frac{dv}{dx} = \frac{v}{x} - \frac{x^2}{\sqrt{v}} \]

### Step 5: Multiply by 2
Multiply the entire equation by 2 to simplify:
\[ \frac{dv}{dx} = \frac{2v}{x} - \frac{2x^2}{\sqrt{v}} \]

### Step 6: Find the Integrating Factor
The standard form of a linear first-order differential equation is:
\[ \frac{dv}{dx} + P(x)v = Q(x) \]
Here, \(P(x) = -\frac{2}{x}\) and \(Q(x) = -2x^2\). The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int P(x) dx} = e^{\int -\frac{2}{x} dx} = e^{-2 \ln |x|} = \frac{1}{x^2} \]

### Step 7: Multiply the Equation by the Integrating Factor
Now we multiply the entire equation by the integrating factor:
\[ \frac{1}{x^2} \frac{dv}{dx} - \frac{2v}{x^3} = -\frac{2}{x^2} \]

### Step 8: Integrate Both Sides
Integrate both sides:
\[ \int \left( \frac{1}{x^2} \frac{dv}{dx} - \frac{2v}{x^3} \right) dx = \int -\frac{2}{x^2} dx \]

### Step 9: Solve the Integrals
The left side becomes:
\[ \frac{v}{x^2} = 2/x + C \]
where \(C\) is the constant of integration.

### Step 10: Substitute Back \(v = y^2\)
Substituting back \(v = y^2\):
\[ \frac{y^2}{x^2} = -2x + C \]
or,
\[ y^2 = -2x^3 + Cx^2 \]

### Final Step: Rearranging the Equation
Rearranging gives us the general solution:
\[ y^2 + 2x^3 + Cx^2 = 0 \]