Task $7$

P$10$

A bridge is designed with a suspension cable that follows a parabolic shape. The equation describing the shape of the cable, in terms of its sag $($vertical distance$)$ as a function of the horizontal distance from the midpoint of the bridge is given by the equation below. $y$ is the vertical displacement and $x$ is the horizontal distance along the bridge:

$\frac{d^{2}y}{d{x}^2}+k^{2}y=0$

Where $k$ is a constant related to the bridge's design.

Assume that the sag of the cable at the midpoint of the bridge is $8$ meters and $k=\frac{1}{4}.$ Find the particular solution for the equation $y$ that satisfies the given condition.

Task $8$

P$10$

The flow of water in an open$-$channel hydraulic system can be described by the differential equation, where $y(x)$ represents the depth of the water at a specific location along the channel, $x.$

$\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$

a$)$ Solve the differential equation above to obtain the general solution for the water depth $y$ in the open channel

b$)$ If initially, the water depth at a certain point in the channel is $y(0)=2m$ and the rate of change of depth is $\frac{dy}{dx}=-0\mathrm{.}5$ meters per minute, determine a particular solution, determine the particular solution for the water depth.

Task $9$

P$10$

The settlement of soil in a foundation is described by the differential equation below, where $y$ represents the vertical settlement at a specific location along the foundation $x.$

$2\frac{d^{2}y}{d{x}^2}+3\frac{dy}{dx}+4y=0$

a$)$ Find the general solution for the depth of settlement

b$)$ For the equation above determine the particular solution for the settlement depth if initially, the settlement depth is $100mm$ when $x=0$ and $\frac{dy}{dx}=-0\mathrm{.}2$ when $x=0$