The general solution is y = y_h + y_p, where y_h is the general solution to the homogeneous equation.
The method of undetermined coefficients is a technique used to find a particular solution to a nonhomogeneous linear differential equation. In this case, we have the differential equation:
y'' - 2y' + y = sin(x)
To use the method of undetermined coefficients, we assume that the particular solution can be written as a linear combination of functions that are similar to the nonhomogeneous term, in this case, sin(x). Since sin(x) is a trigonometric function, we assume that the particular solution is of the form:
y_p = A sin(x) + B cos(x)
where A and B are constants that we need to determine.
Now, we find the first and second derivatives of y_p:
y'_p = A cos(x) - B sin(x)
y''_p = -A sin(x) - B cos(x)
Substituting these derivatives into the original differential equation, we get:
(-A sin(x) - B cos(x)) - 2(A cos(x) - B sin(x)) + (A sin(x) + B cos(x)) = sin(x)
Simplifying, we get:
-2A cos(x) + 2B sin(x) = 0
Since this equation must hold for all x, the coefficients of cos(x) and sin(x) must individually equal zero:
-2A = 0 (1)
2B = 0 (2)
From equation (2), we can solve for B and find that B = 0.
Substituting B = 0 into equation (1), we find that -2A = 0, which implies that A = 0.
Therefore, the particular solution y_p is:
y_p = A sin(x) + B cos(x) = 0
So, the solution to the given differential equation is the sum of the general solution to the homogeneous equation (which we haven't found) and the particular solution:
y = y_h + y_p
where y_h is the general solution to the homogeneous equation.