Substitution for $t^2 y^{\prime \prime}+t y^{\prime}+ y =0$

$t^2 y^{\prime \prime}+t y^{\prime}+ y =0$
Make the substitution $u=\ln t$
Then compute $\frac{dy}{dt}=\frac{dy}{du}\frac{du}{dt}$ and
compute $\frac{d^2y}{dt^2}=\frac{d^2y}{d^2u}(\frac{du}{dt})^2+\frac{dy}{du}\frac{d^2u}{dt^2}$

I'll get the page reference soon. Just wanted to get this up because I said I would.

Phase Portraits Update

As I said in discussion section, instead of using the solutions to determine the directions of the spiral or loop, we can use the original system of equations. Let me know if this is confusing.