you know that when x=1, the value of y=0, also the slope (y') is
flat, and the change of slope (y'') is 0. The only value you
don't know at x=1 is y''', and that can be solved: 3y'''+
5y''-y' +7y=0 ==> 3y'''+ 0 - 0 + 0=0 ==> y''' must equal 0 So,
all given values are 0 and all flat at x=1 ... Now what happens
as you move away from x=1? If y were to change then the slope
(y') will change, and you know that the slope is constrained by
the equation 3y'''+ 5y''-y' +7y=0. So let us say that as x
increases, y increases. So the slope of y goes from 0 to
positive, and so the rate of change in slope (y'') must also
increases, and hence y''' also! So they must all increase
together. What would that do to: 3y'''+ 5y''-y' +7y=0 ? I
suspect (but haven't got as far as proving) that it is not
possible to have y increase, because the various rates of change
would make 3y'''+ 5y''-y' +7y ≠ 0. Sorry, but that is as far as
I have got.