Solve the following differential equations
`ydx+(x-y^3)dy=0`.

Answer

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

### Step 1: Rearranging the Equation
We start by rearranging the given equation:
\[ y \, dx + (x - y^3) \, dy = 0 \]
This can be rewritten as:
\[ (x - y^3) \, dy = -y \, dx \]

### Step 2: Expressing dx/dy
Now, we can express \( \frac{dx}{dy} \):
\[ \frac{dx}{dy} = -\frac{x - y^3}{y} \]
This simplifies to:
\[ \frac{dx}{dy} = -\frac{x}{y} + y^2 \]

### Step 3: Rearranging to Linear Form
Next, we rearrange this into the standard linear form:
\[ \frac{dx}{dy} + \frac{1}{y} x = y^2 \]
Here, we identify \( p = \frac{1}{y} \) and \( q = y^2 \).

### Step 4: Finding the Integrating Factor
To solve this linear differential equation, we need to find the integrating factor \( \mu(y) \):
\[ \mu(y) = e^{\int p \, dy} = e^{\int \frac{1}{y} \, dy} = e^{\log |y|} = |y| \]
Since \( y \) is positive in our context, we can simply use \( \mu(y) = y \).

### Step 5: Multiplying by the Integrating Factor
We multiply the entire differential equation by the integrating factor:
\[ y \frac{dx}{dy} + x = y^3 \]

### Step 6: Recognizing the Left Side as a Derivative
The left-hand side can be recognized as the derivative of a product:
\[ \frac{d}{dy}(xy) = y^3 \]

### Step 7: Integrating Both Sides
Now we integrate both sides:
\[ \int \frac{d}{dy}(xy) \, dy = \int y^3 \, dy \]
This gives:
\[ xy = \frac{y^4}{4} + C \]
where \( C \) is the constant of integration.

### Step 8: Final Solution
Rearranging gives us the final solution:
\[ x = \frac{y^3}{4} + \frac{C}{y} \]

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