Question

X^2y''-7xy'+16y=0 ; y1=x^4
general solution , reduction of order.

Answer 1

Hello,

This exercice is very hard for me. I have found the theory in "Mathematics for students of engineering and applied science" by L.B.Benny (Oxford University Press) page 415"

`x=e^t\\ (dt)/(dx)= (1)/(x)\\ x^2 (d^2y)/(dx^2)= (d^2y)/(dt^2) -(dy)/(dt) \\`


`(d^2y)/(dt^2) - (dy)/(dt) -7 (dy)/(dt) +16y=0\\ (d^2y)/(dt^2) -8 (dy)/(dt) +16y=0\\ \Delta=8^2-4*16=0\\ r=4\\ y=(a+bt)e^(4t) ==\ \textgreater \ y=(a+b\ ln(x))x^4`

If i am wrong, forget all this.