Final answer:
To solve the given differential equation by undetermined coefficients, assume the solution to the homogeneous equation is y_h = A*cos(x) + B*sin(x). Then, guess a particular solution of the form y_p = (Cx + D)*sin(x), and solve for the values of C and D. The general solution is the sum of the homogeneous and particular solutions.
Step-by-step explanation:
To solve the given differential equation by undetermined coefficients, we first assume that the solution to the homogeneous equation y'' + y = 0 is y_h = A*cos(x) + B*sin(x). Next, we look for a particular solution y_p. Since the right-hand side of the equation is of the form f(x)*sin(x), we guess a particular solution of the form y_p = (Cx + D)*sin(x). Plugging this into the differential equation, we can determine the values of C and D.
By substituting the values of C and D into y_p, we get the particular solution y_p = (5/2)*x*sin(x) - x*cos(x). Finally, the general solution to the differential equation is given by adding the homogeneous and particular solutions: y(x) = y_h + y_p. Therefore, the solution to the differential equation y'' + y = 5x*sin(x) is y(x) = A*cos(x) + B*sin(x) + (5/2)*x*sin(x) - x*cos(x), where A and B are constants determined by the initial conditions.