A standard approach to this type of differential equation is to study the homogeneous equation to get its general solution and then to find a particular solution to the original equation.
The homogeneous equation is y'' - 3 y' + 2y =0
After a bit of trial and error, it can be seen that the general solution to this equation is
y = A exp(2x) + B exp(x) {A and B are effectively constants of integration and unknown at this point}
A particular solution to the original equation is y = x/ 2 +5/4 {this is easily checked)
Thus the full solution is y = x/2 + 5/4 + A exp(2x) + B exp(x)
The conditions y(0) =1 and y'(0) =1 are then used to find A and B . I got
A = 5/4 B = -2
