Intro. to Differential Equations

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Discussion Overview

This thread focuses on the topic of Differential Equations, specifically exploring introductory concepts, classifications, and methods for solving first-order differential equations. The discussion includes theoretical aspects, mathematical reasoning, and practical examples, primarily referencing the textbook "Elementary Differential Equations and Boundary Value Problems" by Boyce and DiPrima.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant introduces the concept of Differential Equations, defining them as equations containing a derivative and classifying them into Ordinary Differential Equations (ODE) and Partial Differential Equations (PDE).
  • Examples of ODEs and PDEs are provided, along with distinctions between linear and nonlinear equations.
  • Another participant elaborates on first-order differential equations, presenting the general form and discussing linear equations with variable coefficients.
  • Special cases of first-order equations are discussed, including methods for solving them using integrating factors.
  • Several example problems are presented for practice, illustrating the application of the discussed methods.
  • Participants express a willingness to assist each other with understanding and solving differential equations, indicating a collaborative learning environment.
  • There are mentions of challenges in clarity and presentation, with some participants suggesting improvements for readability and understanding.

Areas of Agreement / Disagreement

Participants generally agree on the definitions and classifications of differential equations, but there are varying levels of understanding and clarity regarding the presentation of solutions and methods. Some participants express confusion about specific steps, indicating that the discussion remains somewhat unresolved in terms of clarity and presentation.

Contextual Notes

Some participants note difficulties in following the mathematical presentation without the textbook, suggesting that assumptions about prior knowledge may not hold for all readers. There are also mentions of missing constants in solutions, which highlight the need for careful attention to detail in mathematical derivations.

Who May Find This Useful

This discussion may be useful for students and individuals interested in learning about differential equations, particularly those seeking collaborative support and clarification on introductory concepts and problem-solving techniques.

  • #91
Erfan said:
I had to solve a first-order nonlinear ODE which led me to a this equation.how can I find the solution for y?
yey=f(x)
Where are the derivatives, e.g., y', or differentials?
 
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  • #92
Erfan said:
I had to solve a first-order nonlinear ODE which led me to a this equation.how can I find the solution for y?
yey=f(x)
"Lambert's W function", W(x), is defined as the inverse function to f(x)= xex. Taking the W function of both sides gives y= W(f(x)).
 
  • #93
So the question should be solved numerically using the Lambert's W function? I mean that can't we then have a function in the form: y=f(x)? or we can no more go further than the Lambert's W function?
 
  • #94
Greg Bernhardt said:
Sounds great! Tutorials like this have been very successful here.

Howto make math symbols:
https://www.physicsforums.com/announcement.php?forumid=73

You can make subscripts and subscripts by using these tags

[ sup ] content [ /sup ]
[ sub ] content [ /sub ]

* no spaces

Mathematicians,
i need an insight and understanding of asymptotic behaviour as applied to singular cauchy problem...anyone can comment...
ken chwala BSC MATHS, MSC APPLIED MATHS FINALIST
 
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  • #95
good job
 
  • #96
Good night,


Last week I begun to study differential equations by my own and first saw ODE's of separable variables. I've learned very well what they are and how to find constant and non-constant solutions. But something extremely trivial is boring me: I can't figure out why some ODE is or is not of separable variable. For example, I know that an ODE of s.v. is an ODE of the type

[; \frac{dx}{dt} = g(t)h(x) ;]​

but I simply cannot say why

[; \frac{dy}{dx}=\frac{y}{x} ;]​

is and ODE of s.v. and why

[; \frac{dy}{dx}=\frac{x+y}{x^2 +1} ;]​

is not.

I know this is very trivial and I am missing something, but I don't know what. Can you help me, please? :-)


[]'s!

Ps.: sorry for my lousy English.
 

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