Final answer:
To solve the given equation, differentiate each term using the product rule and chain rule, simplify the equation by combining like terms, solve for dy/dx, and substitute the given values to find the particular solution.
Step-by-step explanation:
The given equation is (3yx² + 2xy + y³) + (x² + y²)y' = 0. To solve this equation, we need to find the derivative of y with respect to x, denoted as y'. In order to do that, we need to use the product rule and chain rule of differentiation.
Step 1: Differentiate each term of the equation using the product rule and chain rule. For example, the derivative of 3yx² is 3x²(dy/dx) + 6xy. Similarly, differentiate the other terms.
Step 2: Simplify the equation by combining like terms.
Step 3: Solve the equation for dy/dx by isolating the term.
Step 4: The final step is to substitute the given values for x, y, and dy/dx to find the particular solution.