For the differential equation

x$\text{'}=7$x$(1$x$)($x$2)$x$,$

Plot the direction field and superimpose $4$ solutions on the plots. Use the following five initial conditions to obtain these solutions:

x$0=0\mathrm{.}9,$x$0=1\mathrm{.}2,$ x$0=3$ and x$0=2\mathrm{.}5$

Notes:

Please do not modify the already provided code in the cell below

In the code cell follow the $6$ steps outlined below

Step $1$: define a Python function to encode the right hand side of the ODE

Step $2$: define the ODE vector field

Step $3$: normalize the vector field arrows

Step $4$: plot the direction fields

Step $5$: numerically solve the ODE. Each solution corresponds to an Initial Value Problem $($IVP$),$ which consists of the ordinary differential equation and the initial condition. In the cell code below you are asked to use the five initial conditions specified above. These give rise to $4$ solution curves superimposed on the direction field

Step $6$: set graph x and y limits$,$ labels and title.

Use code simular to this provided example:

# MAIN CODE CELL FOR PROBLEM $3$

fig $=$ plt$.$figure$()$

# Step $1.$ Define the right$-$hand side of the ODE $($note the different order$)$

def vf$($x$,$ t$)$:

dx $=$ np$.$zeros$(1)$

dx$[0]=1/($t$**2+$x$[0]**2)$ # dx$[0]$ will change with each problem

return dx

# Step $2.$ Vector field

tgrid$=$np$.$linspace$(-3,5,12)$

xgrid$=$np$.$linspace$(-3,3,12)$

T$,$ X $=$ np$.$meshgrid$($tgrid$,$xgrid$)$

U $=1\mathrm{.}0$

V $=1/($T$**2+$ X$**2)$ # this is the RHS of the ODE. It will change with each problem

# Step $3.$ Normalize arrows

N $=$ np$.$sqrt$($U $**2+$ V $**2)$

U $=$ U $/$ N

V $=$ V $/$ N

# Step $4.$ Use $"$plt$.$quiver$()"$ to plot the direction fields

plt$.$quiver$($T$,$ X$,$ U$,$ V$,$ angles$="$xy$",$ scale$\_$units$=\text{'}$xy$\text{'},$ scale$=7,$ headlength$=0,$headwidth$=1,$color$=$'red'$)$

# Step $5.$ Numerically solve the ODE for various initial conditions $($see below "for x$0$ in $...")$

t$0=-3\mathrm{.}0$

tEnd $=5\mathrm{.}0$

t $=$ np$.$linspace$($t$0,$ tEnd, $100)$

for x$0$ in $[-3\mathrm{.}0,-1\mathrm{.}5,2\mathrm{.}5]$: # you can try other values for x$0$

x$\_$initial $=[$x$0]$

x $=$ integrate.odeint$($vf$,$x$\_$initial, t$)$

plt$.$plot$($t$,$ x$[$:$,0],"-")$

# Step $6.$ Set the plot xlim$,$ ylim$,$ labels and title.

plt$.$xlim$([-3,5])$

plt$.$ylim$([-3,3])$

plt$.$xlabel$(\text{'}$t$\text{'})$

plt$.$ylabel$(\text{'}$x$\text{'})$

figTitle $=$ "Direction fields and solution curves for $$1/($t$^2+$x$^2)$$$"$

plt$.$title$($figTitle$)$

plt$.$show$()$