Final answer:
To solve the initial-value problem, we find the complementary solution by solving the characteristic equation for the homogeneous part, propose a particular solution for the non-homogeneous part, and combine them before applying initial conditions to determine constants.
Step-by-step explanation:
To solve the initial-value problem y" + 6y' - 7y = 11e²⁸, with initial conditions y(0) = 1 and y'(0) = 1, we need to find the particular and complementary solutions of the differential equation.
Step 1: Find the complementary solution (yc).
For the homogeneous part y" + 6y' - 7y = 0, we find the roots of the characteristic equation r² + 6r - 7 = 0. This factors into (r + 7)(r - 1) = 0, giving us r = -7, 1. Therefore, the complementary solution is yc = C1e-7x + C2ex.
Step 2: Find the particular solution (yp).
Since the non-homogeneous part includes the term 11e²⁸, we propose a particular solution of the form yp = Ae²⁸. After differentiating and substituting into the original equation, we solve for A to get the particular solution.
Step 3: Combine yc and yp and apply initial conditions.
Combining the complementary and particular solutions gives us y = yc + yp. We apply the initial conditions to determine the constants C1 and C2 by setting x to 0 and solving the resulting equations y(0) = 1 and y'(0) = 1.