The direction field for

$\frac{dx}{dt}=4t-\frac{2x}{t}=f(t,x)$

$is$ shown below.

$(a)$ For all points given $in$ the table: $(i)$ compute the slope $d\frac{x}{d}tof$ solutions $x(t)$ going through

that point. Highlight the vector with that slope $at$ that point $in$ the direction field. Does the

slope $of$ that vector agree $(roughly)$ with the one you computed?

$(b)$ One can show that $x(t)=t^2+\frac{C}{t^2}is$ a family $of$ solutions $to$ the ODE. Sketch the graphs $ofx(t)$

for $C=0,1,-1$ below, using the method $of$ graphing $by$ superposition $we$ went over $in$ class.

Include the point $on$ the graph with $t=1.$

$C=0,C=1,C=-1$

$(c)$ Add solution curves $(integral$ curves$)to$ the vector field corresponding $toC=0,1,-11,x(1)$