Question

Solve the given differential equation by separation of variables. dy − (y − 1)² dx = 0

Answer 1

To solve the given differential equation dy - (y - 1)^2 dx = 0 by separation of variables, rearrange the equation, integrate both sides, substitute a variable, and solve for the unknowns.

Step-by-step explanation:

To solve the given differential equation by separation of variables, we need to separate the variables and integrate.

1. Rearrange the equation so that terms involving y are on one side and terms involving x are on the other. In this case, it becomes (y-1)^2dy = dx.
2. Integrate both sides with respect to their respective variables. This gives us int((y-1)^2dy) = int(dx).
3. In order to integrate the left side, use the substitution u = y-1. Then the integral becomes int(u^2du).
4. Integrating both sides and simplifying gives us the general solution: (1/3)(y-1)^3 = x + C, where C is the constant of integration.

Therefore, the solution to the given differential equation is (1/3)(y-1)^3 = x + C.