\( y ^ { 1 } - 3 y x ^ { 2 } ( 1 + x ^ { 3 } ) ^ { 3 } = 0 \)

To solve the equation \( y ^ { 1 } - 3 y x ^ { 2} ( 1 + x ^ { 3 } ) ^ { 3 } = 0 \), we can first isolate \( y \). This gives us \( y (1 - 3 x^2 (1 + x^3)^3) = 0 \). This indicates that either \( y = 0 \) or \( 1 - 3 x^2 (1 + x^3)^3 = 0 \). Next, solving \( 1 - 3 x^2 (1 + x^3)^3 = 0 \) involves finding \( x \) values where the expression equals zero, which may require numerical methods or graphing techniques due to its complexity. In practice, understanding how to manipulate and solve such equations can open doors in fields ranging from engineering to finance, especially when dealing with polynomial equations. Many real-world phenomena can be modeled by such equations, making them crucial to grasp. A common mistake when working with polynomials is overlooking solutions that may arise from factoring or manipulating the equation incorrectly. Always check your work, especially if \( y \) can lead to more complex relationships depending on \( x \). A good practice is to verify solutions graphically or through substitution.