Problem $1.[75$ points$]$ Consider the following system of ordinary differential equations:

$\frac{d(vec(x))}{dt}=-$vec$(x)+W(vec(x)),$vec$(x)(0)=$vec$(x{)}_0.$

vec$(x)in{R}^N$ is an $N-$dimensional vector and $Win{R}^{NN}$ is an $NN$ matrix.

The time derivative is applied elementwise. In other words, for vec$(x)=(x_1,x_2,$dots,$x_N)in{R}^N,\frac{d(vec(x))}{dt}=(\frac{d{x}_1}{dt},\frac{d{x}_2}{dt},$dots,$\frac{d{x}_N}{dt}).$

$$ is a hyperbolic tangent function defined as follows. For a scalar zinR,$(z)=tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the standard hyperbolic tangent. For a vector vec$(x)=(x_1,x_2,$dots,$x_N)in{R}^N,$ is applied elementwise. In other words, $(vec(x))=((x_1),(x_2),$dots,$(x_N))in{R}^N.$ Therefore, $W(vec(x))$ is a matrix$-$vector product.

Each element of the matrix $W$ is drawn from a Gaussian distribution with mean $0$ and standard deviation $\frac{g}{\sqrt[\phantom{2}]{N}}.$ In Python, the matrix $W$ can be generated by:

$W=\frac{g}{?}$

$umpy.\sqrt[\phantom{2}]{N}*$ numpy.random.randn $($N$,$N$)$

Each element of the initial condition vec$(x{)}_{0}in{R}^N$ is drawn from a Gaussian distribution with mean $0$ and standard deviation $1.$ In Python, the matrix vec$(x{)}_0$ can be generated by:

$x0=$ numpy.random.randn$($N$)$

In the following problems, find numerical solutions of the ODE under different values of $g,$ using the explicit Euler method with step size $dt=0\mathrm{.}1$ and taking $500$ steps. Choose $N=100,$ i$.$e$.,$ vec$(x)$ is a $100-$dimensional vector.

$(25$ pts$)$ Set up the numerical algorithm $($using Euler's method$)$ that find a numerical solution of the ODE.

$(25$ pts$)$ Let $g=1\mathrm{.}0.$ Find the numerical solution vec$(x)(t)$ of the ODE in the interval tin$[0,50].($a$)$ Plot the first $5$ elements of vec$(x)(t),$ i$.$e$.,x_1(t),$dots,$x_5(t)$ and $($b$)$ describe the behavior of the solution in words.

$(25$ pts$)$ Let $g=2\mathrm{.}0.$ Find the numerical solution vec$(x)(t)$ of the ODE in the interval tin$[0,50].($a$)$ Plot the first $5$ elements of vec$(x)(t),$ i$.$e$.,x_1(t),$dots,$x_5(t)$ and $($b$)$ describe the behavior of the solution in words. Note that the behavior of the solution when $g=1\mathrm{.}0$ and $g=2\mathrm{.}0$ are drastically different.