TT2 #1

[Victor Ivrii]

Solve the following initial value problem
\begin{gather*}
x^3 y''' - 3 x^2 y'' + 6x y'- 6y = - 24 x^{-1} + 6 \ln x \ , \\[4pt]
y(1)= -5/6 \ ,\ y'(1)=-2 \ ,\ y''(1)=2 \ .
\end{gather*}

[Yuan Bian]

homo:  r(r-1)(r-2)-3r(r-1)+6r-6=0
           (r-1)(r-2)(r-3)=0
r₁=1,r₂=2,r₃=3
l(r)=r³-6r²+11r-6
let t=lnx
l(t)=t'''-6t''+11t'-6t=6t-24e^-t
let y_p1=At+B, y_p2=Ce^-t
so Ce^-t(l(-1))=-24e^-t, (l(0)(At+B)+l'(0)A)=6t
C=1, B=-11/6, A=-1
y=c₁x+c₂x²+c₃³+1/x-lnx-11/6
y(1)=−5/6 , y′(1)=−2 , y′′(1)=2
s0 c₁+c₂+c₃+1-11/6=-5/6
    c₁+2c₂+3c₃-2=-2
    2c₂+6c₃+3=2
get c₁=-1/2, c₂=1, c₃=-1/2
y=-x³/2+x²-x/2+1/x-lnx-11/6