The solutions of `y=x((dy)/(dx)+((dy)/(dx))^3)` are given by (where `p=(dy)/(dx)` and k is constant)

Options

A. the constant function ` y=0`
B. `y = p^(-3) e ^(P^2//2)(P +P^3)`
C. `y=(kp^(-3) e^(1//2p^2 ) (P +P^3)`
D. `Ye^(-1 //2)=P^2 +1`

Answer

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

### Step 1: Rewrite the equation using \( p \)
Let \( p = \frac{dy}{dx} \). Then, we can rewrite the equation as:
\[ y = x(p + p^3) \]
This is our equation (1).

### Step 2: Differentiate both sides
Now, we differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}[x(p + p^3)] \]
Using the product rule, we get:
\[ \frac{dy}{dx} = p + x \left( \frac{dp}{dx} + 3p^2 \frac{dp}{dx} \right) \]
This simplifies to:
\[ p = p + x(1 + 3p^2) \frac{dp}{dx} \]

### Step 3: Rearranging the equation
Subtract \( p \) from both sides:
\[ 0 = x(1 + 3p^2) \frac{dp}{dx} \]
This implies:
\[ x(1 + 3p^2) \frac{dp}{dx} = 0 \]
Since \( x \neq 0 \), we can divide by \( x \) (assuming \( x \neq 0 \)):
\[ (1 + 3p^2) \frac{dp}{dx} = 0 \]

### Step 4: Solve for \( p \)
This gives us two cases:
1. \( 1 + 3p^2 = 0 \) which has no real solution.
2. \( \frac{dp}{dx} = 0 \) implies \( p \) is constant.

Let \( p = k \) where \( k \) is a constant.

### Step 5: Substitute back to find \( y \)
Substituting \( p = k \) back into equation (1):
\[ y = x(k + k^3) = xk(1 + k^2) \]

### Step 6: Final form of the solution
Thus, we can express \( y \) as:
\[ y = Cx \quad \text{where } C = k(1 + k^2) \]
This indicates that the solutions are linear in \( x \).

### Conclusion
The final solution of the differential equation is:
\[ y = kx(1 + k^2) \]
where \( k \) is a constant.