PART $2$: Solve all FOUR problems, $10$ points each.

Write a Matlab code to find the root $x$ of the function $ln(1-x)+x+Z**x^2($equated to zero$),$

with $Z=1\mathrm{.}2,1\mathrm{.}3,1\mathrm{.}4,1\mathrm{.}5,1\mathrm{.}6,$ and $1\mathrm{.}7.("ln"$ denotes the natural logarithm.$)$ Use the "Regula

Falsi" method. This equation is quite famous in polymer science: the "Flory Huggins" equation.

When generating the two initial guesses to bracket the root, keep in mind that $x$ is the polymer

mole fraction in a mixture of lots of the polymer and a tiny amount of solvent. Plot the function

versus $x$ in one case showing the root. $($Your code should avoid the trivial root $x=0.$ How?$)$

Check your answers using a Matlab built$-$in function fsolve or fzero $($which one?$)$ instead of

Regula Falsi in each case. Also plot $x($on the $Y$ axis$)$ vs $Z($on the $x$ axis$).$ What insight can you

draw about the role of the parameter $\text{'}Z\text{'}$ from the results?

Use Symbolics in Matlab to find all five roots of $z^5=-32($minus $32$; don't forget the minus

sign!$)$ where, $z$ is a "complex variable." Check your answers by expressing $-1$ via the Euler

Identity and the Euler Formula for complex numbers $($which$,$ incidentally, have no connection

with the Euler Method of numerical solution of differential equations or the Euler equation in

Thermodynamics!$).$ If you don't recall what these terms are, use Google search.

Write a Matlab code that deploys at least FIVE different root$-$finder methods to find the root

near $2$ of the equation $x^3-2x-5=0.$ Print the iterations and the converged result using

format long $($i$.$e$.$ lot more decimals than just $4).$

Use Symbolics in Matlab to solve this Ordinary Differential Equation to find $y$ in terms of $x.$

$\frac{dy}{dx}+ysinx=\frac{1}{2}sin2x.$ Verify your result manually, using the "Integrating Factor" method.