Given the boundary-value problem

$$\begin{array}{ll}{\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}},} & {0<x<1, \quad 0<t<1} \\ {u(0, t)=0,} & {u(1, t)=0, \quad 0 \leq t \leq 1}\end{array}$$

$u(x, 0)=x(1-x),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0, \quad 0 \leq x \leq 1$ use $h=k=\frac{1}{5}$ in equation (6) to compute the values of $u_{i, 1}$ by hand.

Find a formal solution to the vibrating string problem governed by the given initial-boundary value problem. \

$$\frac{\partial^2 u}{\partial t^2}=9 \frac{\partial^2 u}{\partial x^2}, \quad 0<x<\pi, \quad t>0,\\ u(0, t)=u(\pi, t)=0, \quad t>0,\\ u(x, 0)=\sin 4 x+7 \sin 5 x, \quad 0<x<\pi, \\\frac{\partial u}{\partial t}(x, 0)=\left\{\begin{array}{l}x, \quad 0<x<\pi / 2 \\ \pi-x, \quad \pi / 2<x<\pi\end{array}\right.$$

What is the opposite of a partial juror?

Find $\frac{\partial}{\partial x} e^{\sqrt{1+5 y^{-23}}} \ln \left(1-17 y^{97} / z^{47}\right)$

Identify the type of curve for each equation, and then view it on a calculator.

$$4(y^2 + 6y + 1) = x(x - 4) - 24$$

Let T = g(x, y) be the temperature at the point (x, y) on the ellipse $x = 2 \sqrt { 2 } \cos t , \quad y = \sqrt { 2 } \sin t , \quad 0 \leq t \leq 2 \pi,$ and suppose that

$$\frac { \partial T } { \partial x } = y , \quad \frac { \partial T } { \partial y } = x.$$

Suppose that T = xy - 2. Find the maximum and minimum values of T on the ellipse.

Find a formal solution to the given initialboundary value problem.

$$\begin{aligned} &\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}, \quad 0<x<\pi, \quad t>0 \\ &u(0, t)=0, \quad u(\pi, t)=3 \pi, \quad t>0 \\ &u(x, 0)=0, \quad 0<x<\pi \end{aligned}$$

In view of Clapeyron's equation and figure, is there something special about ice $I$ versus the other forms of ice ?

Determine the partial derivative with respect to x and the partial derivative with respect to y at x=4 and y=1 for $f(x, y)=3x^2y^3+2x+3y^6+9$.

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. Show that a solution of the BVP $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, \quad-\infty<x<\infty, \quad 0<y<1$ $\left.\frac{\partial u}{\partial y}\right|_{y=0}=0, \quad u(x, 1)=f(x), \quad-\infty<x<\infty$ is $u(x, y)=\frac{1}{\pi} \int_{0}^{\infty} \int_{-\infty}^{\infty} f(t) \frac{\cosh \alpha y \cos \alpha(t-x)}{\cosh \alpha} d t d \alpha$.

Sketch a magnified view (showing atoms/molecules) of each of the following and explain: a homogeneous mixture of an element and a compound

Show that $\frac{\partial(G / T)}{\partial(1 / T)}=H$. Chemists measure the enthalpy of a reaction by measuring this rate of change.

Use a proportion to answer the question. What percent of 70 is 21?

View at least two cycles of the graphs of the given functions on a calculator. $y=\frac{1}{2} \sec 3 x$