(1) Solve the following initial value problems: (a) uy = xuux, u(x, 0) = x (b) xux +
yuy + uz = u, u(x, y, 0) = h(x, y).

To solve these initial value problems, we use the method of characteristics, which is often employed for solving first-order partial differential equations (PDEs). Let's tackle each problem separately.

### (a) Problem:
\[
u_y = x u u_x, \quad u(x, 0) = x
\]

#### Step 1: Rewrite the PDE in a characteristic form

Given:
\[
u_y = x u u_x
\]

Let's rewrite this equation using the characteristic equations. The characteristic equations for this PDE can be derived as:

\[
\frac{dx}{1} = \frac{dy}{xu} = \frac{du}{x u u_x}
\]

This set of equations comes from matching the coefficients of \(dx\), \(dy\), and \(du\).

#### Step 2: Solve the characteristic equations

From \(\frac{dx}{1} = \frac{dy}{xu}\), we have:

\[
\frac{dy}{dx} = x u \implies y = \frac{x^2 u}{2} + C_1
\]

Now, solving \( \frac{dx}{1} = \frac{du}{x u u_x} \):

Since \(u\) is a function of \(x\), and let's simplify this system, we plug in \( u(x, 0) = x \).

Using the initial condition, we get \(u(x,0) = x\).

Thus the solution is a combination of the expressions from each.

### (b) Problem:
\[
x u_x + y u_y + u_z = u, \quad u(x, y, 0) = h(x, y)
\]

#### Step 1: Rewrite the PDE in a characteristic form

Given:
\[
x u_x + y u_y + u_z = u
\]

The characteristic equations can be written as:

\[
\frac{dx}{x} = \frac{dy}{y} = \frac{dz}{1} = \frac{du}{u}
\]

#### Step 2: Solve the characteristic equations

From \(\frac{dx}{x} = \frac{dy}{y} = \frac{dz}{1}\):

1. Solving the first two:
   
   \[
   \frac{dx}{x} = \frac{dy}{y} \implies \ln x - \ln y = C_1 \implies \frac{x}{y} = e^{C_1}
   \]

   Let this constant \(e^{C_1}\) be \(C_1\), so \( x = C_1 y \).

2. Solving \(\frac{dz}{1}\):

   \(z = C_2 \), from \(u = h \).

Combining these, find \(u\).

Would you like to go deeper into each step or have additional problems related to this? 

### Related Questions:

1. How do you find solutions using characteristics?
2. What makes PDEs differ from ODEs in characteristic applications?
3. Can any PDE be solved analytically, or do some require numerical solutions?
4. How would changing initial conditions affect the problem?
5. Are there scenarios where the method of characteristics fails?

### Tip: Ensure you match conditions when checking solutions!

Solve the following initial value problems: (a) uy = xuux, u(x, 0) = x (b) xux + yuy + uz = u, u(x, y, 0) = h(x, y).

Explore step-by-step solutions to initial value problems involving partial differential equations using the method of characteristics. Learn to solve PDEs like uy = xuux and xux + yuy + uz = u with detailed explanations suitable for advanced calculus students.

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