$1.($a$)$ Find the solution of the following system of nonlinear

equations.

$xy+x^2-y^3-1=0$

$x+2y-x{y}^2-2=0$

$($b$)$ Write a concise computer program $($using either MATLAB or

PYTHON$)$ to implement this.

$2.($a$)$ A function is provided as mentioned below:

$f(x)=1+2x^2$

Evaluate the integral $($ I $)$ numerically in a range between $0$

and $2.$

$I={\displaystyle \underset{0}{\overset{2}{}}}f(x)dx$

$($b$)$ Write a computer program $($using either MATLAB or

PYTHON$)$ to implement the numerical integration

$($considering the number of discretizations as $n).$

A beam $($having support at both ends$)$ of a length $(L)$ of $10$ m

is subjected to some loading such that deflection in vertical

direction $($denoted as $w)$ is expressed using the following

differential equation:

$\frac{d^{2}w}{d{x}^2}=f(x)$

where, $x$ represents the position along the length of the

beam. The $f(x)$ is known: $f(x)=0$ at $x=0$ and $x=L,f(x)=100$

unit at $x=\frac{L}{2},$ having linear variation of $f(x)$ elsewhere.

Besides, it can be assumed that $w=0$ at $x=0$ and $x=L.$

Evaluate the deflections at $\frac{L}{4},\frac{L}{2},$ and $3\frac{L}{4}$ numerically

using the Finite Difference Method.