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Charpit’s Method Charpit’s Method or General Method to Solve Non-Linear Partial Differential Equation Charpit’s Method: It is a well-known mathematical technique known as “Method of Characteristics”. The method is widely used for solving first-order or quasi-linear PDE’s. Explanation of Method: Let the given differential equation be: Different Types of Solutions in PDE \[ f(x, y, z, p, q) = 0 \quad \text{(i)} \quad \text{where} \quad z = z(x, y) \] We know \( dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy \). [Therefore, z=z(x,y)] \( dz = p \, dx + q \, dy \quad \text{(ii)} \). Now we shall find another relation \( F(x, y, z, p, q) = 0 \quad \text{(iii)} \) so that on solving equations (i) and (iii) for \( p \) and \( q \) and putting these values in (ii), equation (ii) becomes integrable, and this integral gives the complete solution of (i) (integral). For finding \( F \), differentiate (i) with resp...

In Partial Differential Equation questions are solving to some important forms Lagrange's Linear Equation, Charpit's Method and etc. Here the concept of solutions are very vital to understand for solving the P.D Equation. Types of Solutions There are four types of solutions are listed below - Complete Solution Particular Solution Singular Solution General Solution Complete Solution When a partial differential equation f(x,y,z,p,q) = 0      ...(i) where, z is the function of independent variables x and  y and f   be the function of dependent variables. Here we obtain a relation g(x,t,z,a,b) = 0 there are many arbitary constants as a number of independent variables, then g(x,y,z,p,q) = 0 is called the complete solution or integral of eq( i ) for example, z = ax + by +f(a,b) is the complete solution of z = px +qy +f(p,q) Particular Solution First of all we have a differential equatio...