Final answer:
The given differential equation is not exact.
Step-by-step explanation:
The equation provided is x * dy/dx = 5xe^x - y + 9x^2. To determine if it is exact, we will check if it satisfies the condition for exactness: ∂M/∂y = ∂N/∂x, where M and N are the respective coefficients of dx and dy.
Firstly, let's rearrange the given equation into a more standard form: y' = (5xe^x - y + 9x^2)/x, where y' = dy/dx. Upon examination, M = ∂M/∂y = -1 and N = ∂N/∂x = 5e^x + 18x. Since ∂M/∂y ≠ ∂N/∂x, the equation is not exact.
To find an integrating factor to make it exact, we'll use an integrating factor μ. Here, μ is a function of x such that μMdx + μNdy becomes an exact differential equation. To determine μ, we'll use the formula: μ = e^(∫((∂N/∂x - ∂M/∂y)/M)dx).
Upon computation, μ = e^(∫((5e^x + 18x + 1)/x)dx). Unfortunately, this equation doesn't yield an easily integrable form. Therefore, a straightforward integrating factor cannot be obtained, and the equation remains not exact.
Hence, without an integrating factor to make the equation exact, it is not feasible to solve this differential equation using the standard exact differential equation methods