Final answer:
The solution to the differential equation dy/dx = y² - y³ involves separation of variables, partial fraction decomposition, and integration, resulting in an implicit solution with logarithms.
Step-by-step explanation:
You have presented the first-order nonlinear differential equation dy/dx = y² - y³. To solve this, we can employ a method called separation of variables. By treating the differentials as fractions (which is permissible under certain circumstances), we can separate the y's to one side and the x's to the other.
First, we rearrange the equation to form:
dy / (y² - y³) = dx
Next, factor the denominator on the left side:
dy / y²(1 - y) = dx
We can then integrate both sides. The left side will be a partial fraction decomposition, and the right side is a standard integral in terms of x.
The final solution will involve an implicit form with logarithms and possibly an arbitrary constant, which represents the family of possible solutions to the differential equation.