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))
