#x(d^2y)/dx^2-4xdy/dx+4xy=5xe^(3x)+4xe^(2x)#
Eliminate common #x# term
#(d^2y)/dx^2-4dy/dx+4y=5e^(3x)+4e^(2x)#
Expressing in terms of linear differential operator #D = d/(dx)#:
#(D^2-4D+4)y=5e^(3x)+4e^(2x)#
Factoring:
#(D-2)underbrace((D-2)y)_(= z)=5e^(3x)+4e^(2x) qquad triangle#
#z' - 2 z=5e^(3x)+4e^(2x)#
Integrating factor: #exp(int (-2) dx) = e^(- 2x)#
Distribute integrating factor across equation:
# (e^(-2x) z)^' =5e^x+4 #
Integrating:
# e^(-2x) z =5e^x+4x + c_1#
# :. z =5e^(3x)+4xe^(2x) + c_1e^(2x) qquad bb(= (D-2)y)#
#y'-2 y = 5e^(3x)+4xe^(2x) + c_1e^(2x) #
Using same integrating factor again:
#(e^(-2x)y)^' = 5e^( x)+4x + c_1 #
Integrating:
# e^(-2x)y = 5e^( x)+ 2x^2 + c_1 x + c_2#
#y = 5e^( 3x)+ 2x^2e^(2x) + c_1 xe^(2x) + c_2e^(2x)#
# = e^(2x)( 5e^x+ 2x^2 + c_1 x + c_2 )#
