Final answer:
The question involves solving a boundary-value problem for a second-order linear differential equation with given boundary conditions. The solution process includes finding the general solution and applying the provided boundary value y(0) = 0. A second well-defined boundary condition is needed to determine the exact solution.
Step-by-step explanation:
The question requires solving a boundary-value problem in differential equations, which includes a second-order linear differential equation and boundary conditions. The known values are given by y(0) = 0 and the equation y" - 2y' + 2y = 2x - 2. To solve the problem, we must find the general solution to the differential equation and then apply the boundary conditions to determine the specific solution that satisfies both conditions.
First, find the complementary solution by solving the homogeneous equation y" - 2y' + 2y = 0. Afterward, determine a particular solution to the non-homogeneous equation using methods such as undetermined coefficients or variation of parameters. Once the general solution is obtained, apply the boundary condition y(0) = 0 to find the constants involved in the solution.
Unfortunately, without the second boundary condition (y(?) = ?) being properly defined, we cannot find the exact solution. If the second boundary condition were provided, we would substitute it into the general solution to find the remaining constants and hence the specific solution to the boundary-value problem.