Final answer:
The general solution of the differential equation 2x² d²y/dx² - 6x dy/dx + 6y = 2 is obtained by finding the homogeneous solution and a particular solution using the method of undetermined coefficients and summing them.
Step-by-step explanation:
To find the general solution for the differential equation 2x² d²y/dx² - 6x dy/dx + 6y = 2, we follow the method of undetermined coefficients. First, we need to find the corresponding homogeneous equation by setting the right-hand side to zero:
2x² d²y/dx² - 6x dy/dx + 6y = 0
The solutions to this homogeneous equation are typically of the form x^n, where n is a constant that satisfies the characteristic equation.
Outside of solving that, the next step is to construct a particular solution to the non-homogeneous equation by assuming a form for y that is based on the right-hand side, in this case, a constant. From this, coefficients are determined that make the assumed solution satisfy the given equation.
After both the general solution of the homogeneous equation and a particular solution of the non-homogeneous equation have been found, the final general solution is the sum of these two solutions.