The given Cauchy problem is:

$u_x + u_y = u^2, \quad u(x, 0) = x^2.$

Step 1: Setting Up the Method of Characteristics

To solve this first-order partial differential equation using the method of characteristics, we start by rewriting the equation in the following form:

$\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = u^2.$

We now use the method of characteristics, which involves finding curves along which the solution is constant. The characteristics are defined by the system of equations:

$\frac{dx}{ds} = 1, \quad \frac{dy}{ds} = 1, \quad \frac{du}{ds} = u^2.$

Here, $s$ is the parameter along the characteristic curves.

Step 2: Solving the Characteristic Equations

1. Solving for $x(s)$ and $y(s)$:

From the system $\frac{dx}{ds} = 1$ and $\frac{dy}{ds} = 1$, we integrate with respect to $s$:

$x(s) = s + x_0 \quad \text{and} \quad y(s) = s + y_0,$ where $x_0$ and $y_0$ are the initial conditions.

2. Solving for $u(s)$:

Now, we solve the equation $\frac{du}{ds} = u^2$. This is a separable differential equation, so we can write:

$\frac{du}{u^2} = ds.$

Integrating both sides:

$-\frac{1}{u} = s + C,$ where $C$ is a constant of integration. Solving for $u(s)$, we get:

$u(s) = \frac{1}{C - s}.$

Step 3: Using the Initial Condition

The initial condition is given as $u(x, 0) = x^2$. At $y = 0$, we have $s = 0$, so:

$u(x_0, 0) = \frac{1}{C} = x_0^2.$

Thus, $C = \frac{1}{x_0^2}$.

Step 4: Expressing the Solution

Now that we have the value of $C$, we can write the solution for $u(s)$:

$u(s) = \frac{1}{\frac{1}{x_0^2} - s}.$

Using the relationship $x(s) = s + x_0$ and $y(s) = s + y_0$, we express $s$ in terms of $x$ and $y$. Specifically, from $x = s + x_0$, we have $s = x - x_0$. Therefore, the solution becomes:

$u(x, y) = \frac{1}{\frac{1}{x_0^2} - (x - x_0)}.$

This gives the solution of the PDE in terms of the initial condition.

Final Solution:

The final form of the solution is:

$u(x, y) = \frac{1}{\frac{1}{x^2} - (x - x_0)}.$

Conclusion

This method provides the general solution to the Cauchy problem. Would you like more details on any of the steps or need help with another part of the problem?