Answer:
The general solution is

+ 
Step-by-step explanation:
Step :1:-
Given differential equation y(4) − 2y''' + y'' = e^x + 1
The differential operator form of the given differential equation
comparing f(D)y = e^ x+1
The auxiliary equation (A.E) f(m) = 0




The roots are m=0,0 and m =1,1
complementary function is 
<u>Step 2</u>:-
The particular equation is 
P.I = 
P.I = 
P.I = 



applying in integration u v formula

= 





again integration 
The general solution is 

+ 