Final answer:
To determine if the equation is exact, we check if the partial derivatives of the terms with respect to x and y are equal. The equation is exact, so we find the integrating factor and solve the equation. The solution is 3y⁴ - 3y⁶/2 + 3x/2 y³ = C.
Step-by-step explanation:
To determine whether the equation is exact, we need to check if the partial derivatives of the terms with respect to x and y are equal. Let's find the partial derivatives of the given equation:
Partial derivative with respect to x: (∂M/∂x) = 9/y
Partial derivative with respect to y: (∂N/∂y) = -9y + (9x/y²)
If (∂M/∂x) = (∂N/∂y), then the equation is exact. In this case, (∂M/∂x) = (∂N/∂y), so the equation is exact. To solve the exact equation, we can find an integrating factor. Let's find the integrating factor:
Multiply the equation by y² to get:
9y dx - (9y³ - 9x) dy = 0
The integrating factor is the coefficient of dy which is y². Multiply the entire equation by y²:
9y³ dx - (9y⁵ - 9xy²) dy = 0
The equation is now exact. To find the solution, we can integrate the terms separately. The solution is given by:
∫(9y³ dx) - ∫(9y⁵ - 9xy²) dy = C
Simplifying and integrating, we get:
3y⁴ - 3y⁶/2 + 3x/2 y³ = C