Differential Equations

About Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives. 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.
I’ve 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 wasn’t 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 can’t anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that I’ve 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: Euler’s 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