The system of equations is $x^2y''=(x+1)y+xz$ and $x^2z''=-(x+1)z-xy$
I don't want the full answer, I just have no idea how to start. The techniques we were given were all for second order things and I'm not sure how to extend it to systems. If there is some trick with the equations let me know.
Also, the books says very explicitly not to approximate x+1 by x
You make a system of four equations out of your two equations.
$u_1=y, u_2=y', u_3=z, u_4=z'$
The system would be in terms of $u$ and $u'$ so you can linearize it.