Find the temperature u(x,t) in a bar of silver of length 10 cm and constant cross section of area 1 cm² (density 10.6 g/cm³, thermal conductivity 1.04 cal/(cm sec °C), specific heat 0.056 cal/(g °C) that is perfectly insulated laterally, with ends kept at temperatures 0°C and initial temperature f(x)°C, where f(x)=4-0.8|x-5|

System consisting of a cylindrical copper rod insulated on its lateral surface while its ends are in contact with hot and cold walls at temperatures $1000^\circ \text{R} \hspace{1mm} \text{and} \hspace{1mm} 500^\circ \text{R}$ ,respectively

$$\begin{array}{l l } \text{(a) Sketch the variation of temperature with position through the rod, x.}\\ \text{(b) Is the rod in equilibrium? Explain.}\\ \end{array}$$

A laterally insulated bar of length 10cm and constant cross-sectional area $1 \mathrm{~cm}^2$, of density $10.6 \mathrm{gm} / \mathrm{cm}^3$, thermal conductivity $1.04 \mathrm{cal} /\left(\mathrm{cm} \sec { }^{\circ} \mathrm{C}\right)$, and specific heat $0.056 \mathrm{cal} /\left(\mathrm{gm}^{\circ} \mathrm{C}\right.$ ) (this corresponds to silver, a good heat conductor) has initial temperature $f(x)$ and is kept at $0^{\circ} \mathrm{C}$ at the ends $x=0$ and $x=10$. Find the temperature $u(x, t)$ at later times. Here, $f(x)$ equals:

$f(x)=\sin 0.1 \pi x+\frac{1}{2} \sin 0.2 \pi x$

A laterally insulated bar of length 10cm and constant cross-sectional area $1 \mathrm{~cm}^2$, of density $10.6 \mathrm{gm} / \mathrm{cm}^3$, thermal conductivity $1.04 \mathrm{cal} /\left(\mathrm{cm} \sec { }^{\circ} \mathrm{C}\right)$, and specific heat $0.056 \mathrm{cal} /\left(\mathrm{gm}^{\circ} \mathrm{C}\right.$ ) (this corresponds to silver, a good heat conductor) has initial temperature $f(x)$ and is kept at $0^{\circ} \mathrm{C}$ at the ends $x=0$ and $x=10$. Find the temperature $u(x, t)$ at later times. Here, $f(x)$ equals:

$f(x)=1-0.2|x-5|$

Find the steady-state oscillations of y''+cy'+y=r(t) with c>0 and r(t) as given. Note that the spring constants is k=1. Show the details. r(t)=t(π²-t²) if -π<t<π and r(t+2π)=r(t)

Implement $f(x, y, z)=\Sigma m(0,1,3,4,7)$ as a two-level gate circuit, using a minimum number of gates. (a) Use AND gates and NAND gates. (b) Use NAND gates only.

Find the steady-state oscillation of $y^{\prime \prime}+c y^{\prime}+y=r(t)$ with $c>0$ and r(t) as given. (Show the details of your work.)

$r(t)=\sin 3 t$

The heat flux of a solution $u(x, t)$ across $x=0$ is defined by $\phi(t)=-K u_x(0, t)$. Find $\phi(t)$ for the solution . Explain the name. Is it physically understandable that $\phi$ goes to 0 as $t \rightarrow x$ ?

Consider the following matrices. Find the permutation matrix P so that PA can be factored into the product LU, where L is lower triangular with 1s on its diagonal and U is upper triangular for these matrices.

Consider the Bessel equation of order v $t^{2} y^{\prime \prime}+t y^{\prime}+\left(t^{2}-v^{2}\right) y=0$ where v is real and positive. (a) Find a power series solution $J_{\nu}(t)=\frac{t^{\nu}}{2^{\nu} \nu !} \sum_{n=0}^{\infty} a_{n} t^{n}, \quad a_{0}=1.$ This function $J_v(t)$ is called the Bessel function of order v. (b) Find a second solution if 2v is not an integer

Consider

$$\left(1-x^2\right) y^{\prime \prime}-2 x y^{\prime}+\left[n(n+1)-\frac{m^2}{1-x^2}\right] y=0 .$$

Substituting $y(x)=\left(1-x^2\right)^{m / 2} v(x)$, show that $v$ satisfies

$$\left(1-x^2\right) v^{\prime \prime}-2(m+1) x v^{\prime}+[n(n+1)-m(m+1)] v=0 . \tag{15}$$

Starting from $\left(1-x^2\right) y^{\prime \prime}-2 x y^{\prime}+n(n+1) y=0$ and differentiating it $m$ times, show that a solution of $(15)$ is

$$v=\frac{d^m P_n}{d x^m} .$$

The corresponding $y(x)$ is denoted by $P_n{ }^m(x)$. It is called an associated Legendre function and plays a role in quantum physics. Thus

$$P_n{ }^m(x)=\left(1-x^2\right)^{m / 2} \frac{d^m P_n}{d x^m}$$

Find the Fourier transform of $f(x)$ . Show the details.

$$f(x)=\left\{\begin{aligned} e^{2 i x} & \text { if }-1<x<1 \\ 0 & \text { otherwise }\end{aligned}\right.$$

In this problem:

$$\left(4-x^2\right) y^{\prime \prime}+2 y=0, \quad x_0=0$$

a. Seek power series solutions of the given differential equation about the given point $x_0$; find the recurrence relation that the coefficients must satisfy.

b. Find the first four nonzero terms in each of two solutions $y_1$ and $y_2$ (unless the series terminates sooner).

c. By evaluating the Wronskian $W\left[y_1, y_2\right]\left(x_0\right)$, show that $y_1$ and $y_2$ form a fundamental set of solutions.

d. If possible, find the general term in each solution.

The boundary-value problem $y^{\prime \prime}=y^{\prime}+2 y+\cos x, \quad 0 \leq x \leq \frac{\pi}{2}, \quad y(0)=-0.3, \quad y\left(\frac{\pi}{2}\right)=-0.1$ has the solution $y(x)=-\frac{1}{10}(\sin x+3 \cos x).$ Use the Linear Shooting method to approximate the solution, and compare the results to the actual solution. a. With $h=\frac{\pi}{4}$ b. With $h=\frac{\pi}{8}$