Auxiliary equation is
Now P.I. =
then substitute x= o in last both result and solve for
Yes, and Kjchauhan did that rather "offhandedly" with " ". Quite a jump!
You can solve for the particular solution by "variation of parameters" or, more simply in this case, "undetermined coefficients". The right hand side of the equation is so normally, we would try something of the form . But we have already seen that was a solution to the associated homogeneous equation so there will be no value of A that will give anything but "0" on the left.
Instead try . Then and . Putting those into the equation gives so we can take .
That gives as kjchauhan said.
Of course, you are still not finished. Now find and so that y(0)= 8/9 and y'(0)= 1.