Final answer:
The general solution to the differential equation dx = x⁷ y dy is obtained through separation of variables and integration, resulting in the solution -1/6x⁶ = 1/2 y² + C, where C is the constant of integration.
Step-by-step explanation:
The differential equation dx = x⁷ y dy can be approached by separating variables, which involves rearranging the equation so that each variable and its differential are on opposite sides of the equation. This yields dx/x⁷ = y dy. Integrating both sides, we get:
Integral of 1/x⁷ dx = Integral of y dy
Resulting in -1/6x⁶ = 1/2 y² + C, where C is the constant of integration.
Thus, the general solution to the differential equation is -1/6x⁶ = 1/2 y² + C.