To solve the first-order partial differential equation \( p(p^2 + 1) + (b - z)q = 0 \) using the **Charpit's method**, we follow these steps. Charpit's method is used for solving non-linear first-order PDEs of the form \( F(x, y, z, p, q) = 0 \), where \( p = \frac{\partial z}{\partial x} \) and \( q = \frac{\partial z}{\partial y} \).

Given:
\[
F = p(p^2 + 1) + (b - z)q = 0
\]

### Step 1: Identify Partial Derivatives of \( F \)

1. \( F = p(p^2 + 1) + (b - z)q \).
2. Calculate partial derivatives with respect to \( x \), \( y \), \( z \), \( p \), and \( q \).

\[
\frac{\partial F}{\partial z} = -q
\]
\[
\frac{\partial F}{\partial p} = 3p^2 + 1
\]
\[
\frac{\partial F}{\partial q} = b - z
\]
Since \( F \) does not depend on \( x \) and \( y \) explicitly:
\[
\frac{\partial F}{\partial x} = 0, \quad \frac{\partial F}{\partial y} = 0
\]

### Step 2: Charpit's Auxiliary Equations

The Charpit's auxiliary equations are:
\[
\frac{dx}{\frac{\partial F}{\partial p}} = \frac{dy}{\frac{\partial F}{\partial q}} = \frac{dz}{p \frac{\partial F}{\partial p} + q \frac{\partial F}{\partial q}} = \frac{dp}{- \frac{\partial F}{\partial x} - p \frac{\partial F}{\partial z}} = \frac{dq}{- \frac{\partial F}{\partial y} - q \frac{\partial F}{\partial z}}
\]

Substituting our values:
\[
\frac{dx}{3p^2 + 1} = \frac{dy}{b - z} = \frac{dz}{p(3p^2 + 1) + q(b - z)} = \frac{dp}{0 + qp} = \frac{dq}{0 + pq}
\]

### Step 3: Simplify and Solve the System

These equations can now be solved step-by-step using integration techniques. Each equation provides a path to build the complete solution by integrating across

Explore a detailed solution to the partial differential equation p(p^2 + 1) + (b - z)q = 0 using Charpit's method. This problem delves into non-linear PDE concepts and utilizes Charpit’s auxiliary equations for advanced calculus students.

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