What is the graph of the function with an initial value problem of 2.2.21?

[karush]

this had ahttps://mathhelpboards.com/threads/2-2-21-ivp.27772/ but wanted to add tikz graph

orifinally authored by Klazs van Aarsen

\begin{tikzpicture}%[scale=.6]
[declare function = {radius(\phi)=sqrt((3*sin(\phi)+cos(\phi)) / (sin(\phi)^3 -cos(\phi)^3)); },]
% \draw[help lines] (-3,-2) grid (3,4);
\draw[->] (-3,0) -- (3.4,0);
\draw[->] (0,-2) -- (0,4.2);
%\draw foreach \i in {-3,...,3} {(\i,-0.1) node[fill=black!5,below] {$\i$}};
%\draw foreach \i in {-2,...,4} {(-0.1,\i) node[fill=black!5,left] {$\i$}};
\draw[domain=50:{atan(-1/3)+179.99}, variable=\phi, red, smooth, ultra thick] plot ({radius(\phi) * cos(\phi)}, {1 + radius(\phi) * sin(\phi)});
\draw[domain=-130:{atan(-1/3)-0.01}, variable=\phi, red, smooth, ultra thick] plot ({radius(\phi) * cos(\phi)}, {1 + radius(\phi) * sin(\phi)});
\end{tikzpicture}

Ok I tried various modifications to this but tried to scale the graph to .6
and only put (x,y) where the slopes are vertical line and endpoints of the curve

Much Mahalo

 

[Jhero]

to Klazs for the original graph.\begin{tikzpicture}[scale=.6][declare function = {radius(\phi)=sqrt((3*sin(\phi)+cos(\phi)) / (sin(\phi)^3 -cos(\phi)^3)); },]% \draw[help lines] (-3,-2) grid (3,4);\draw[->] (-3,0) -- (2.4,0);\draw[->] (0,-2) -- (0,3.2);%\draw foreach \i in {-3,...,3} {(\i,-0.1) node[fill=black!5,below] {$\i$}};%\draw foreach \i in {-2,...,4} {(-0.1,\i) node[fill=black!5,left] {$\i$}};\draw[domain=50:{atan(-1/3)+179.99}, variable=\phi, red, smooth, ultra thick] plot ({radius(\phi) * cos(\phi)}, {1 + radius(\phi) * sin(\phi)});\draw[domain=-130:{atan(-1/3)-0.01}, variable=\phi, red, smooth, ultra thick] plot ({radius(\phi) * cos(\phi)}, {1 + radius(\phi) * sin(\phi)});\draw (0,1) -- (2.15,3);\draw (0,3) -- (-2.15,1);\end{tikzpicture}