y'''' -6y'''+9y''=x
i take y_p=ax+b
i get 0=x
so i cant find the a ,b for the solution
Your characteristic equation is
$\displaystyle r^4-6r^3+9r^2=0 \iff r^2(r-3)^2$
so the complimentary solution already contains $\displaystyle y=ax+b $
Since there is a factor of $\displaystyle r^2$ you must multiply your guess by $\displaystyle x^2$ to get $\displaystyle ax^3+bx^2$
what are they laws tieing the homogeneus and the practical solutions?
i we have also (r-3)^2 thing
why its not changing the particular solution too
?
whats the general theory on think link thing
?
i dont know how to search it on google
Try this link, it may help.
The Method of Undetermined Coefficients
Also try google Method of undetermined coeffients.