Final answer:
The general solution of the higher-order differential equation y''' - 4y'' - 5y' = 0 is y(t) = C1 + C2e^(5t) + C3e^(-t), where C1, C2, and C3 are constants determined by the initial conditions.
Step-by-step explanation:
Finding the General Solution of a Higher-Order Differential Equation
To find the general solution of the third-order differential equation y′′′ − 4y′′ − 5y′ = 0, we first assume a solution of the form y = ert, where r is a constant to be determined. Plugging this into the equation, we get the characteristic equation r3 - 4r2 - 5r = 0. Factoring this gives us r(r2 - 4r - 5) = r(r - 5)(r + 1) = 0, which has roots r = 0, r = 5, and r = -1.
Therefore, the general solution to the differential equation is y(t) = C1 + C2e5t + C3e-t, where C1, C2, and C3 are arbitrary constants determined by initial conditions.
This solution covers a family of functions that satisfy the given third-order differential equation, and we used algebra skills to factor and solve for the roots of the characteristic polynomial to find the explicit form of these functions.