Let F={y(x)=C1+C2x+C3ex∣C1,C2,C3∈R}. A class is investigating whether F constitutes the general solution to the third-order differential equation y′′′−y′′=0. Which of the following provides the best justification for the conclusion?
a) Because the characteristic equation has a repeated root, one of the constants is lost; therefore only two arbitrary constants should appear, so F is too large to be the general solution.
b) Substituting y into the differential equation yields −C1−C2x=0, forcing C1=C2=0; hence F cannot represent the general solution.
c) The family F contains three linearly independent arbitrary constants—matching the third order of the differential equation—and every member satisfies y′′′−y′′=0, so it is the general solution.
d) Because ex already includes the series terms 1 and x, the constants on 1 and x are redundant; at least four arbitrary constants are required for a fully general solution.