is given by:

$tanx-x=0$

The bulking load $(P_c)$ is given as: $(E\frac{\text{I}}{L^2})x^2,$ where $L$ is the length of the column, $E$ is Young's modulus of elasticity and I is the moment of inertia of the cross$-$section.

$1\mathrm{.}1$ Without the use of Excel or any computerised tool, estimate the least load $($it should not be zero$),$ using Newton's method, that the column can carry without buckling. Your starting point as $x_o=4\mathrm{.}3.$

Note: The specific values of the parameters are not given, therefore, you must express your answer in terms of these parameters.

$1\mathrm{.}2$ Now consider this column to be subjected to a time$-$varying axial load $P(t),$ the lateral deflection $y(x,t)$ of the column can be described by the following second$-$order differential equation:

$EI(d^2\frac{y}{d}{x}^2)+P(t)y=0$

where: $y$ is the lateral displacement, $x$ is the position along the column length, t is time and the applied load $P(t)=P_0(1-e^{-t}).$

Given the following initial conditions:

$3$

Fixed at base: $y(0,t)=0$

Free end: $y^{\text{'}}(L,t)=0$

Initial deformation: $y(x,0)=0\mathrm{.}01L**sin(\frac{x}{2L})$

Initially at rest: $d\frac{y}{d}t(x,0)=0$

Use the fourth$-$order Runge$-$Kutta $($RK$4)$ method with time step $h=0\mathrm{.}01s,$ column length $L=2m,El=2000N*m^2,P_0=1000N$ and $=0\mathrm{.}5s^{-1}$ to solve for the lateral displacement $y(L,t)$ at the free end of the column for $t=0$ to $t=5$ seconds. Show all your steps.

$1\mathrm{.}3$ Note that the system can be converted to a system of first$-$order ODEs. Write and solve this system of equations. Report the maximum displacement to four decimal places.

$1\mathrm{.}4$ Write a code that can be used to solve Question $1\mathrm{.}3,$ this code should consider the conversion of the system to first$-$order ODE. You do not need to print the results of the code, but it should be functional. Explain what each command seeks to achieve.

OR

Explain in full how you would solve this on Excel. Your explanations should be detailed and clear.

$(10)$