Final answer:
To solve the given initial-value problem: (x y)² dx + (2xy x² - 5) dy = 0, y(1) = 1, we can use the method of exact differential equations.
Step-by-step explanation:
To solve the given initial-value problem: (x y)² dx + (2xy x² - 5) dy = 0, y(1) = 1, we can use the method of exact differential equations.
To determine if the equation is exact, we check if the partial derivative of the coefficient of dx with respect to y is equal to the partial derivative of the coefficient of dy with respect to x. In this case, the partial derivatives are 2xy and (2x - 5), respectively.
Since the partial derivatives are not equal, we can't solve the equation as an exact differential equation. Therefore, this method won't work for this problem. It may be necessary to consider other methods or techniques to solve this initial-value problem.