The characteristic equation of the homogeneous equation (d^2y/dx^2) - 5(dy/dx) + 8y = 0 is:
r^2 - 5r + 8 = 0
The roots of this equation are r1 = 2 and r2 = 4.
Therefore, the general solution to the homogeneous equation is:
y_h = c1e^(2x) + c2e^(4x)
Now, we need to find a particular solution to the non-homogeneous equation.
Since the right-hand side of the equation is xex, which is a product of a polynomial and an exponential function, we can assume a particular solution of the form:
y_p = (Ax + B)ex
Taking the first and second derivatives of y_p:
y_p' = Aex + (Ax + B)ex
y_p'' = 2Aex + (Ax + B)ex
Substituting these expressions into the differential equation:
2Aex + (Ax + B)ex - 5(Aex + (Ax + B)ex) + 8(Ax + B)ex = xex
Simplifying:
(-3A + 8B)ex = xex
Therefore, we have:
-3A + 8B = x
To solve for A and B, we differentiate y_p:
y_p' = Aex + (Ax + B)ex
y_p'(0) = A + B = 0
Therefore, B = -A.
Substituting this into the equation -3A + 8B = x, we get:
-3A + 8(-A) = x
Solving for A, we get:
A = -x/5
Substituting this into B = -A, we get:
B = x/5
Therefore, the particular solution to the differential equation is:
y_p = (-x/5 + x/5)ex = (0)ex = 0
The general solution to the non-homogeneous equation is:
y = y_h + y_p = c1e^(2x) + c2e^(4x)
Therefore, the general solution to the differential equation is:
y = c1e^(2x) + c2e^(4x)