Characteristic equation of differential equation is: #r^3-3r^2+2r=0# or #r*(r-1)*(r-2)=0#
Its roots are #r_1=0#, #r_2=1# and #r_3=2#
Consequently homogeneous solution of it,
#y_h=c_1+c_2*e^x+c_3*e^(2x)#
I use variation of parameters for particular solution of it,
#y_p=u_1*1+u_2*e^x+u_3*e^(2x)=u_1+u_2*e^x+u_3*e^(2x)#
Now, I have to solve these equation system,
#(u_1)'*1+(u_2)'*e^x+(u_3)'*e^(2x)=0# or #(u_1)'+(u_2)'*e^x+(u_3)'*e^(2x)=0# #(1)#
#(u_1)'*0+(u_2)'*e^x+(u_3)'*2e^(2x)=0# or #(u_2)'*e^x+2(u_3)'*e^(2x)=0# #(2)#
#(u_1)'*0+(u_2)'*e^x+(u_3)'*4e^(2x)=e^(2x)/(e^x+1)# or #(u_2)'*e^x+4(u_3)'*e^(2x)=e^(2x)/(e^x+1)# #(3)#
#[(u_2)'*e^x+4(u_3)'*e^(2x)]-[(u_2)'*e^x+2(u_3)'*e^(2x)]=e^(2x)/(e^x+1)-0#
#2(u_3)'*e^(2x)=e^(2x)/(e^x+1)#
#(u_3)'=1/2*1/(e^x+1)#
#u_3=1/2int (dx)/(e^x+1)#
=#1/2int (e^(-x)*dx)/(e^(-x)+1)#
=#1/2Ln(e^(-x)+1)#
=#1/2Ln((e^x+1)/e^x)#
=#1/2Ln(e^x+1)-1/2Ln(e^x)#
=#1/2Ln(e^x+1)-x/2#
Consequently,
#(u_2)'*e^x+2(u_3)'*e^(2x)=0#
#(u_2)'=-2e^x*(u_3)'#
#(u_2)'=-2e^x*1/2*1/(e^x+1)#
#(u_2)'=-e^x/(e^x+1)#
#u_2=-Ln(e^x+1)#
Hence,
#(u_1)'-e^x/(e^x+1)*e^x+1/2*1/(e^x+1)*e^(2x)=0#
#(u_1)'-e^(2x)/(e^x+1)+1/2*e^(2x)/(e^x+1)=0#
#(u_1)'=1/2*e^(2x)/(e^x+1)#
#u_1=1/2int (e^(2x)*dx)/(e^x+1)#
=#1/2 int [(e^x+1)*(e^x-1)+1]/(e^x+1)*dx#
=#1/2 int (e^x-1)*dx#+#1/2 int (dx)/(e^x+1)#
=#1/2*(e^x-x)+1/2*(Ln(e^x+1)-x)#
=#1/2e^x-x/2+1/2Ln(e^x+1)-x/2#
=#1/2e^x+1/2Ln(e^x+1)-x#
Hence,
#y_p=u_1+u_2*e^x+u_3*e^(2x)#
=#1/2e^x+1/2Ln(e^x+1)-x-e^xLn(e^x+1)+1/2e^(2x)*Ln(e^x+1)-x/2*e^(2x)#
=#1/2e^x-1/2Ln(e^x-1)*(2e^x-1)-1/2e^(2x)(x-Ln(e^x+1))#
Thus,
#y=y_h+y_p#
=#c_1+c_2e^x+c_3e^(2x)+1/2e^x-1/2Ln(e^x-1)*(2e^x-1)-1/2e^(2x)(x-Ln(e^x+1))#
