Solve the following differential equations.
`dy //dx + y //x = y^3`

Answer

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ \frac{dy}{dx} + \frac{y}{x} = y^3 \]
This can be rearranged to:
\[ \frac{dy}{dx} = y^3 - \frac{y}{x} \]

### Step 2: Identify the integrating factor
The equation can be expressed in the standard form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \( P(x) = \frac{1}{x} \) and \( Q(x) = y^3 \).

To find the integrating factor \( \mu(x) \), we use the formula:
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\log x} = x \]

### Step 3: Multiply through by the integrating factor
Now, we multiply the entire differential equation by the integrating factor \( x \):
\[ x \frac{dy}{dx} + y = xy^3 \]

### Step 4: Rewrite the left-hand side
The left-hand side can be rewritten as the derivative of a product:
\[ \frac{d}{dx}(xy) = xy^3 \]

### Step 5: Integrate both sides
Now, we integrate both sides:
\[ \int \frac{d}{dx}(xy) \, dx = \int xy^3 \, dx \]
This gives us:
\[ xy = \int xy^3 \, dx + C \]
where \( C \) is the constant of integration.

### Step 6: Solve for \( y \)
To find \( y \), we will need to solve the integral on the right-hand side. However, we can express the general solution as:
\[ xy = \frac{x^2}{2} y^3 + C \]

### Step 7: Rearranging the equation
Rearranging gives us:
\[ xy - \frac{x^2}{2} y^3 = C \]

### Final Solution
This is the implicit solution of the differential equation:
\[ xy - \frac{x^2}{2} y^3 = C \]