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.

Page 421 Problem 15

Page 421 Problem 15

Phase Portraits

Alright, so here's a visual aid regarding phase portraits. I included most cases. I omitted the case when one of the eigenvalues is zero. Hopefully your notes on that case are clear.

You can (and probably should) click on the scanned pages to make them bigger (there are two pages).