Find the type, transform to normal form, and solve. Show your work in detail.

$$u_{xx}-6u_{xy}+9u_{yy}=0$$

Find the type, transform to normal form, and solve. Show your work in detail.

$$u_{xx}-4u_{xy}+5u_{yy}=0$$

Familiarize yourself with parametric representations of important surfaces by deriving a representation, by finding the parameter curves (curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. xy-plane r(u, v)=(u, v)(thus ui+vj)

Find the type, transform to normal form, and solve. Show your work in detail.

$$u_{xx}+2u_{xy}+10u_{yy}=0$$

Solve the equation $u_x + 2xy^2u_y = 0$.

Find $u_{x x}^{\prime \prime}$ when $u$ and $v$ are defined as functions of $x$ and $y$ by the equations $x y+u v=1$ and $x u+y v=0$.

Suppose the system

$$\left\{\begin{array}{l}{x u + y v-u v=0} \\ {y u-x v+u v=0}\end{array}\right.$$

can be solved for u and v in terms of x and y, so that u = u(x, y) and v = v(x, y). Use implicit differentiation to find the partial derivatives $\frac{\partial u}{\partial x}$ and $\frac{\partial v}{\partial x}$.

How does the frequency of the eigenfunctions of the rectangular membrane change (a) If we double the tension? (b) If we take a membrane of half the density of the original one? (c) If we double the sides of the membrane? Give reasons.

If a steel wire 2 m in length weighs 0.9 nt (about 0.20 lb) and is stretched by a tensile force of 300 nt (about 67.4 lb), what is the corresponding speed of transverse waves?

State three or four of the most important PDEs and their main applications.

Find the deflection u(x, t) of the string of length L=π and c²=1 for zero initial displacement and "triangular" initial velocity u_t(x, 0)=0.01x if 0≤x≤½π, u_t(x, 0)=0.01(π-x) if ½π≤x≤π. (Initial conditions with u_t(x, 0)≠0 are hard to realize experimentally.)

Find u(x, t) fr the string of length L=1 and c²=1 when the initial velocity is zero and the initial deflection with small k(say, 0.01) is as follows. Sketch or graph u(x, t). 2x-4x² if 0<x<½, 0 if ½<x<1

Find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it.

$$y'=1/(6e^y-2x)$$

Find u(x, t) for the string of length :=1 and c²=1 when the initial velocity is zero and the inital deflection with small k(say, 0.01) is as follows. Sketch or graph u(x, t). kx²(1-x)