Final answer:
To solve the initial value problem y'' - 8y' + 16y = 0 with y'(0) = 1/8, assume a solution of the form y = e^(rt). Solve the characteristic equation to find the repeated roots. Use the initial condition to find the specific solution.
Step-by-step explanation:
To solve the initial value problem y'' - 8y' + 16y = 0 with y'(0) = 1/8, we can assume a solution of the form y = e^(rt). Substituting this into the equation gives the characteristic equation r^2 - 8r + 16 = 0. Solving this quadratic equation, we find two repeated roots: r = 4.
Using the repeated root, the general solution is y = (c1 + c2t)e^(4t), where c1 and c2 are constants. To find the specific solution, we use the initial condition y'(0) = 1/8. Taking the derivative of y and setting t = 0, we get y'(0) = c1 - 4c2 = 1/8.
Solving this system of equations, we find c1 = 1/8 and c2 = 0. Therefore, the solution to the initial value problem is y = (1/8)e^(4t).