Final answer:
The differential equation y'' - y = 0 has a general solution of the form y = C1e^x + C2e^-x. This solution is obtained by solving the characteristic equation r^2 - 1 = 0 with roots r = 1 and r = -1.
Step-by-step explanation:
Solving the Differential Equation
The differential equation y'' - y = 0 represents a second-order linear homogeneous differential equation. To solve this equation, we will look for solutions that can take the form of exponentials, erx. Assuming a solution of the form y = erx, we substitute this into the equation to obtain a characteristic equation of the form r2 - 1 = 0. The roots of this equation are r = 1 and r = -1. Hence, the general solution to the differential equation is y = C1ex + C2e-x, where C1 and C2 are constants determined by initial conditions.
To verify that this is a solution, we can differentiate y to get y' = C1ex - C2e-x, and a second time to obtain y'' = C1ex + C2e-x. Substituting y'' and y into the original equation, we will find that the equation holds true for these forms of y, confirming that our general solution is correct.