downlaod

YouTube Video Downloader

YouTube MP3/MP4 Downloader


Comments

Post a Comment

Popular posts from this blog

Liouville's Theorem in Complex Analysis

Image
Liouville's Theorem Art-15. Liouville's Theorem This theorem is very important, if you read this theorem carefully then some of the theorems coming up are based on this on which you can apply it and there are many questions where you can apply this theorem and you can solve those questions with Liouville's Theorem . Statement: If a function \( f(z) \) is analytic for all finite values of \( z \) and is bounded, then \( f(z) \) is constant. Proof: Let \( z_1, z_2 \) be any two points of the \( z \)-plane. Draw a circle \( C \) with center at the origin and radius \( R \), enclosing the points \( z_1 \) and \( z_2 \) so that: \( |z_1| < R \) and \( |z_2| < R \) Since \( f(z) \) is bounded, there exists \( M > 0 \) such that \( |f(z)| \leq M \) on \( C \). By Cauchy's integral formula, we have \( f(z_1) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_1} \, dz \) and \( f(...
Read more

Charpit’s Method or General Method to Solve Non-Linear Partial Differential Equation

Image
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...
Read more