Given an IVP

an$($x$)$dnydxn$+$an$1($x$)$dn$1$ydxn$1+...+$a$1($x$)$dydx$+$a$0($x$)$y$=$g$($x$)$

y$($x$0)=$y$0,$y$($x$0)=$y$1,,$y$($n$1)($x$0)=$yn$1$

If the coefficients an$($x$),...,$a$0($x$)$ and the right hand side of the equation g$($x$)$ are continuous on an interval I and if an$($x$)0$ on I then the IVP has a unique solution for the point x$0$ in I that exists on the whole interval I.Consider the IVP on the whole real line

$($x$2+25)$d$4$ydx$4+$x$4$d$3$ydx$3+1($x$225)$dydx$+$y$=$sin$($x$)$

y$(11)=84,$y$(11)=17,$y$(11)=4,$y$(11)=10,$

The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a