4.4.1. Charpit’s Method (for Non-linear Equations)

Given: F(x, y,z, p,q)=0 We find complete solution by solving the Charpit's equations: dx d y dz d p dq = = = = F F pF +qF −F −pF −F −qF p q p q x z y z

Charpit's Method is used to solve non-linear first-order PDEs. The starting point is a function defined as F(x, y, z, p, q) = 0, where p and q are the partial derivatives of z with respect to x and y, respectively. This equation represents a relationship where multiple independent variables are connected through the dependent variable z. To find a complete solution using Charpit's method, we need to solve the Charpit's equations, which express relationships between differentials (dx, dy, dz) and the derivatives (dp, dq).

Think of Charpit's method like navigating a complex network of roads (the PDEs) using a set of maps (the Charpit's equations). Each road (solution) connects locations (variables) in a unique way, and by solving these equations, we can find the most efficient paths (solutions) through the network.

Charpit’s equations are expressed as: dx d y dz d p dq = = = = F F pF +qF −F −pF −F −qF p q p q x z y z

Charpit's equations consist of a set of relationships that describe how changes in the independent variables (x, y) affect changes in the dependent variable (z) and how these are interconnected through their derivatives (p, q). The notation implies that for a complete solution to the PDE, we need to analyze these interconnections. Each part of the equation corresponds to a different aspect of how the variables influence each other, with terms involving derivatives highlighting the roles of p and q in the relationships. This structured approach helps in systematically deriving the complete solution.

Imagine you are learning how to cook. Each ingredient (the variables) affects the final dish (the solution). Charpit's equations tell you how each ingredient must be combined and adjusted (the relationships between the variables) to achieve the perfect flavor (the complete solution).

📌 Example 3: Solve p2 +q2 =1 Apply Charpit’s equations and solve for x,y,z.

In this example, we are required to solve the non-linear PDE p² + q² = 1 using Charpit's method. To do this, we set up Charpit's equations based on our given function F, which is defined by the equation p² + q² – 1 = 0. The next step involves deriving the necessary relationships (dx, dy, dz, dp, dq) from the original function using the relationships defined in Charpit's equations, thereby allowing us to unravel the solution for x, y, and z based on the connections established between p and q.

Consider a situation where you're trying to maintain a balanced diet (the condition p² + q² = 1), with p representing one type of nutrient and q representing another. To find out how much of each nutrient you need (the solutions for x, y, z), you can refer to a set of guidelines (Charpit’s equations) that help you understand the relationships between food quantities and nutrients, ultimately leading you to a balanced meal.

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