solving (x+a)^2 d2y/dx2 -4(x+a) dy/dx + 6y = x

Nov 2019
1
0
India
Hi, my question is that can we find the particular integral of this kind of equation by variation of parameters? If yes, then what mistake did I make in my solution in the provided link? If this is not the method to adopt, what other method can we use?
 
Dec 2015
972
128
Earth
\(\displaystyle (x+a)^2 y'' -4(x+a) y' + 6y = x\).
Set \(\displaystyle x+a=e^{u}\).
 
Dec 2015
972
128
Earth
H-homogeneous solution , p-particular solution .
\(\displaystyle H’’-5H’+6H=0\) ; \(\displaystyle H=c_1 e^{2u } +c_2 e^{3u}.\)

\(\displaystyle \begin{cases}p=v_1 (u)e^{2u} +v_2 (u) e^{3u} \\ p’’-5p’+6p-x=0 \end{cases} \)

Solve the system of equations for \(\displaystyle v_1 , v_2 \).
The general solution is \(\displaystyle y=H+p\).