Final answer:
The differential equation y'' + 4y = 4sin(2x) can be solved with undetermined coefficients by finding a particular solution that includes the x multiplied by cosine and sine functions. The general solution includes terms from the homogeneous equation and the -1/4xcos(2x) from the particular solution, with constants determined by initial conditions.
Step-by-step explanation:
To solve the given differential equation by the method of undetermined coefficients, we need to find a particular solution to the non-homogeneous differential equation y'' + 4y = 4sin(2x). Since the right-hand side is a sinusoidal function, for the particular solution we choose a function of the same form: Axsin(2x) + Bxcos(2x).
We use this form because the homogeneous solution already contains the terms sin(2x) and cos(2x), and the presence of x accounts for this.
After substituting the particular solution and its derivatives into the differential equation, equating coefficients, and solving for A and B, we find that the particular solution involves a term like -1/4xcos(2x).
Therefore, the general solution of the differential equation is y(x) = C2 sin(2x) + C1 cos(2x) - 1/4xcos(2x), where C1 and C2 are constants that are determined by initial conditions.