Question $2(50$ pts$)-$ The following equations are used to analyze uniform beams with distributed loads:

$\frac{dy}{dx}=(x)$;$\frac{d}{dx}=\frac{M(x)}{EI}$;$\frac{dM}{dx}=V(x)$;$\frac{dV}{dx}=-w(x)$

The equation for rotation versus position is given below:

$(x)=\frac{w_0}{120\text{E}\text{I}\text{L}}(-5x^4+6L^2{x}^2-L^4)$

$E=190$GPa

$I=0\mathrm{.}0004m^4$

$w_0=3\mathrm{.}0k\frac{N}{m}$

$L=4m$

Assume that the deflection is at $x=0$ and at $x=L.$

Using Simpson's $\frac{1}{3}$ method when $x=1m$ and $x=0\mathrm{.}5m$ :

$$ Obtain numerical values of deflection $($y $($m$))$ along the beam $($from $0$ to $4m)$

How does $x($step size$)$ impact the answer?

Use Simpson's $3/8$ method to estimate deflection $($y $($m$))$ along the beam when $x=1\mathrm{.}33m.$

Use the central, backward, or forward difference equations $($depending on the points available$)$

and find $M(Nm)$ values along the beam when $x=1m.($i$.$e$.,$ at $x=0,x=1,x=2,x=3,$ and $x=4)$