Final answer:
To find the general solution to the given differential equation with a known particular solution y1 = e^x, employ the reduction of order technique to find another linearly independent solution, and express the general solution as a combination of both.
Step-by-step explanation:
The question involves finding the general solution to a second-order linear homogeneous differential equation with variable coefficients, xy'' - (x + 1)y' + y = 0, where it is given that y1 = e^x is a particular solution. To find the complete general solution, one must find a second linearly independent solution and combine it with the given particular solution. This process usually involves employing the reduction of order technique, where one assumes that the second solution has the form v(x)e^x, and then substitutes into the differential equation to solve for v(x). After finding v(x), the second solution will be of the form v(x)e^x, and the general solution to the differential equation can be expressed as y = C1e^x + C2v(x)e^x, where C1 and C2 are arbitrary constants determined by initial conditions or boundary conditions.