Final answer:
The representative family of solutions for the differential equation y' = y - 1 is given by option C, y = Ce^x - 1. This solution, when differentiated, yields the correct form that satisfies the original equation, with the constant C determining the specific solution within the family.
Step-by-step explanation:
To sketch a representative family of solutions for the differential equation y' = y − 1, we first need to find the general solution. We are looking for a function whose derivative is equal to the function itself minus one. Analyzing the given options, we can determine that option C, y = Cex − 1, is the correct solution. This is because when we differentiate y with respect to x, we get y' = Cex which simplifies to y' = y + 1, fitting the original equation.
The other options do not give us the proper form when differentiated. For example, option A, y = Ce(x+1), would lead to y' = Ce(x+1) and does not simplify to y' = y − 1. Option B and D also do not satisfy the equation upon differentiation.
Thus, the family of solutions would include curves that are effectively the exponential function shifted down by one unit, giving us a horizontal asymptote at y = -1, and the constant C will determine the particular solution within that family, representing various y-intercepts.