This document provides an overview of mathematical methods in physics, including partial differential equations and differential equations. It discusses important second-order partial differential equations like the wave equation, diffusion equation, and Poisson's equation. It also covers general solutions to first and second order differential equations, separation of variables, integral transform methods, and Green's functions. Examples are provided of solving separable, homogeneous, and exact first order differential equations, as well as Bernoulli, Riccati, and other nonlinear equations.

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Mathematical Methods in Physics
210 Partial Differential Equations: Introduction to important PDEs in Physics (wave equation, diffusion
equation, Poisson’s equation, Schrodinger’s equation), general form of solution, general and particular
solutions (first order, inhomogeneous, second order), characteristics and existence of solutions,
uniqueness of solutions, separation of variables in Cartesian coordinates, superposition of separated
solutions, separation of variables in curvilinear coordinates, integral transform methods, Green’s

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functions
Differential Equation
Example
Important Second-Order PDEs

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Here c is a positive constant, t is time, x, y, z are Cartesian coordinates, and dimension
is the number of these coordinates in the equation.

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Example
Separable first order equations
The differential equation

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Homogeneous differential equations
The first order ODE in differential form
Exact equations
The first order ODE

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Test for exactness
The differential equation

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The Bernoulli equation is a nonlinear first order differential equation with the standard
form
Reduces
to the linear first order ODE
The Riccati equation is an important nonlinear equation with the standard form

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Example
Example

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Example
Example

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Example
Solve the equation

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Example

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Example

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Example

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Example

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Example
Example

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Example

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Example

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Example

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Example

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