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In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives.[1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions.
Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers.
The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
Here inside App are my notes for my differential equations course .
Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn how to solve differential equations or needing a refresher on differential equations.
Ive tried to make these notes as self-contained as possible and so all the information needed to read through them is either from a Calculus or Algebra class or contained in other sections of the notes.
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Because I wanted to make this a fairly complete set of notes for anyone wanting to learn differential equations have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here.
You will need to find one of your fellow class mates to see if there is something in these notes that wasnt covered in class.
Sometimes questions in class will lead down paths that are not covered here.
I try to anticipate as many of the questions as possible when writing these up, but the reality is that I cant anticipate all the questions.
Sometimes a very good question gets asked in class that leads to insights that Ive not included here.
You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.
App Differential Equations Include :
differential equations
stochastic differential equations
partial differential equations
integral math way
Slope fields
Linear equations and the integrating factor
Autonomous equations
Numerical methods: Eulers method
numerical analysis data science
numerical methods for engineers solutions
integral method
numerical analysis in machine learning
solve the equation for exact solutions over the interval
Linear PDE
Mechanical Vibrations
Nonhomogeneous Equations
Higher order linear ODEs (Exercises)
Matrices and linear systems
Eigenvalue Method
Multiple Eigenvalues
Matrix exponentials
Vectors, Mappings, and Matrices
Matrix Algebra
Elimination
Subspaces, Dimension, and The Kernel
Inner Product and Projections
math way linear algebra
Table of Laplace Transforms
Power Series
Series Solutions of Linear Second Order ODEs
Power series methods (Exercises)
Steady Periodic Solutions
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