Final answer:
The solution to the given differential equation (d⁵y)/(dx⁵)-(d³y)/(dx³)=0 is y = Ae^(-x) + Be^(0x) + Ce^x, where A, B, and C are constants.
Step-by-step explanation:
To solve the differential equation (d⁵y)/(dx⁵)-(d³y)/(dx³)=0, we can assume the solution is of the form y = e^(kx), where k is a constant. By substituting this into the differential equation and simplifying, we get (k⁵-k³)e^(kx) = 0. To satisfy this equation, either k⁵-k³=0 or e^(kx) = 0. The second case is not possible, so solving the first equation gives us three possible values for k: k = -1, 0, or +1. Therefore, the general solution to the differential equation is y = Ae^(-x) + Be^(0x) + Ce^x, where A, B, and C are constants.