given that the solution is Y=ln x * v(x)
Find V(x)
2 Answers

xy'' + y' = 0
y1 = lnx
Y = v(x) * y1
Y = v ln(x) ( i wanna make it simpler but keeping in mind that v is a function of x)
dY/dx = v (1/x) + ln(x) *dV/dx
Y' = v/x + v'ln(x)
Y'' = 2v'/x  v/x² + v''ln(x)
replace in the original diff equa; xy'' + y' = 0
xy'' + y' = 0
x{2v'/x  v/x² + v''ln(x) } + v/x + v'ln(x) = 0
2v'  v/x + xv''ln(x) + v/x + v'ln(x) = 0
2v' + xv''ln(x) + v'ln(x) = 0
let u = v'
```````````````` keeping in mind that u = u(x)
u' = v''
2v' + xv''ln(x) + v'ln(x) = 0
2u + xu' ln(x) + uln(x) = 0
xu'lnx = ulnx  2u
`````````````````````````` separable
u' (xlnx) = u(2 + lnx)
u' / u = (2 + lnx) / (xlnx)
(1/u) du/dx = (2 + lnx) / (xlnx)
(1/u) du = (2 + lnx) / (xlnx) dx
integrate both sides to get
∫ (1/u) du = ∫ (2 + lnx) / (xlnx) dx
u = C / (xln²x)
but u = v'
so
v(x) = A + B/lnx

Like whitesox suggested, there comes a factor the position you purely can't get a closed form answer. The (3x^2)^(x^2) is amazingly difficult to combine. yet the position did a majority of these human beings come from. you do no longer have any contacts.