Final answer:
To solve the given homogeneous initial-value problem (x² - 2y²) dx + dy = xy, y(-1) = 3, a substitution like y = vx is used to simplify the equation. After separation and integration, the initial condition is applied to find the particular solution.
Step-by-step explanation:
The question involves solving an initial-value problem with a given differential equation that is homogeneous. The differential equation provided is (x² - 2y²) dx + dy = xy, and the initial condition is y(-1) = 3. We approach this problem by searching for an appropriate substitution that simplifies the equation, commonly using variables that reveal the homogeneity of the equation.
To solve the homogeneous differential equation, we would typically use a substitution like y = vx or y/x = v, which transforms the original equation into one with variables x and v only. After substituting and simplifying, we can separate variables or maybe even find a direct integration path. The end result would provide us with the general solution, which we can then use together with the initial condition to find the particular solution.
Finally, applying the initial condition y(-1) = 3 allows us to determine the constants in the general solution, giving us the particular solution to the initial-value problem.