To solve the first-order Partial Differential Equation (PDE) $p^2 + q + uu = 0$, using the method of characteristics, we follow these steps systematically.

Step 1: Rewrite the PDE

The given PDE is:

$p^2 + q + u u = 0$

where:

• $p = \frac{\partial u}{\partial x}$
• $q = \frac{\partial u}{\partial y}$

Substitute the expressions for $p$ and $q$:

$\left(\frac{\partial u}{\partial x}\right)^2 + \frac{\partial u}{\partial y} + u u = 0$

This is a nonlinear PDE, and we aim to use the method of characteristics to solve it.

Step 2: Characteristic Equations

The characteristic equations for a PDE of the form $F(x, y, u, p, q) = 0$ are derived from the following system:

$\frac{dx}{ds} = \frac{\partial F}{\partial p}, \quad \frac{dy}{ds} = \frac{\partial F}{\partial q}, \quad \frac{du}{ds} = p\frac{\partial F}{\partial p} + q\frac{\partial F}{\partial q}$

For the given PDE, $F(p, q, u) = p^2 + q + u u$, we can differentiate it with respect to $p$ and $q$ to find the characteristic equations.

1. From $\frac{dx}{ds} = \frac{\partial F}{\partial p}$, we get: $\frac{dx}{ds} = 2p$

2. From $\frac{dy}{ds} = \frac{\partial F}{\partial q}$, we get: $\frac{dy}{ds} = 1$

3. From $\frac{du}{ds} = p \frac{\partial F}{\partial p} + q \frac{\partial F}{\partial q}$, we get the equation for $u$:

$\frac{du}{ds} = 2p \cdot p + 1 \cdot 1 = 2p^2 + 1$

Now, we have the system of characteristic equations:

$\frac{dx}{ds} = 2p$ $\frac{dy}{ds} = 1$ $\frac{du}{ds} = 2p^2 + 1$

Step 3: Solving the Characteristic System

1. Solve $\frac{dy}{ds} = 1$:

From $\frac{dy}{ds} = 1$, we get:

$y = s + C_1$

where $C_1$ is a constant.

2. Solve $\frac{dx}{ds} = 2p$:

Next, we solve $\frac{dx}{ds} = 2p$. To do so, we need an expression for $p$. Recall that $p = \frac{\partial u}{\partial x}$, but for now, let's keep solving the system step-by-step.

3. Solve $\frac{du}{ds} = 2p^2 + 1$:

We solve $\frac{du}{ds} = 2p^2 + 1$ similarly by integrating once we find a relation for $p$.

Step 4: Integrating for the Solution

After solving the system of characteristics, we would express $u(x, y)$ in terms of $x$, $y$, and the constants of integration. However, due to the nonlinearity of the PDE, further steps would involve manipulating the system to derive explicit forms for the characteristics, which often leads to parametric solutions depending on initial conditions or boundary data.

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Would you like me to go further into the specifics of solving the characteristic equations or clarify any step?

Follow-up Questions:

1. How do you solve the characteristic system explicitly?
2. What are the initial conditions or boundary values required for this problem?
3. How do you interpret the method of characteristics geometrically?
4. Can this method be applied to higher-order nonlinear PDEs?
5. What are some typical applications of the method of characteristics in physics?

Tip: When using the method of characteristics, always look for simplifications in the characteristic system that might reduce the complexity of the problem.