I am only given y= x/3 and am asked to find the general solution. So what I did was get the derivative up until the second order and I multiplied each so that if I were to add them up they would equal zero. The question I have is...how do you choose what to multiply the derivatives by? Because I took a quiz and the first time I did they had x^2y''+7xy'+9y=0 and the second time they had switched it to x^2y''+3xy-3y=0.
Please excuse my lack of clarity. I was up all night doing DE. Now here is a link to a very similar problem. http://i9.photobucket.com/albums/a92...dadsadsaad.jpg
Let the ED equation be...
(1)
... and we suppose to know a particular solution of it... in this case. A general procedure to find a second solution independent from the first is illustrated now...
If u and v are both solutions of (1) is...
(2)
Multiplying the first of (2) by v, the second by u and taking the difference we obtain...
(3)
Setting the (3) is written as...
(4)
... and its silution is...
(5)
Deviding both termes of (4) by and taking into account (5) we have...
(6)
... the solution of which is...
(7)
If we observe (7) is easy to see that if v is solution of (1), then...
(8)
... is also solution of (1). In particular...
(9)
... so that the general solution of (1) is...
(10)
Kind regards