Ordrinary Differential Equation Problem

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Homework Help Overview

The problem involves solving an ordinary differential equation (ODE) given by y'(x) = cos[y(x)-x]. The original poster discusses various methods attempted, including expressing the ODE in complex form and using trigonometric identities, but expresses difficulty in finding a way to separate variables or identify an alternative solution method.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants suggest a substitution method, u(x) = y(x) - x, as a potential approach. There is also discussion about the importance of recognizing simpler methods before attempting more complex ones. Questions are raised about how to develop problem-solving skills for ODEs and the reasoning behind choosing specific methods.

Discussion Status

The discussion is ongoing, with some participants providing guidance on recognizing simpler approaches and reflecting on the learning process. There is an acknowledgment of the challenges faced by those less experienced with ODEs, and a focus on developing problem-solving techniques through practice.

Contextual Notes

Some participants note the original poster's limited experience with ODEs, which may influence their approach to the problem. There is an emphasis on the need to explore different methods and the potential for overlooking simpler solutions.

Aequiveri
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The problem asks me to solve the following ordinary differential equation for y(x):

y'(x) = cos[y(x)-x]

I have tried multiple methods to solve this equation, including expressing the O.D.E. in its complex form. I also tried expanding the O.D.E. using the angle difference identity

cos[y(x)-x] = cos[y(x)]cos[x]+sin[y(x)]sin[x]

However, I can't seem to find a way to separate the variables nor do I see an alternative method to solve this O.D.E.

Any help you can give to lead me in the right direction would be greatly appreciated. Thanks!
 
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How about substituting u(x)=y(x)-x?
 
Sorry, that was a lot simpler than I thought. Thanks for your help!
 
Before you put this out of your mind... how would you come up with that on your own? What about this problem should suggest you try something like that?

Remember, you don't just want to be able to solve this problem -- you want to be able to solve other problems you haven't seen yet, so you want to distill this lesson into a tool you can use in the future.
 
Thanks for the advice. I haven't had much experience with ODEs, so I think the best way to learn problem-solving techniques involving ODEs might be to practice problems over and over again until the appropriate methods involved in obtaining solutions becomes more obvious.
 
Sure. I think you were just overlooking the 'simple' before you launched into the 'hard' ways of solving it. Wrong order.
 
Well, this one can be tricky for some people -- there are two "obvious" ways to attack the problem via simplifying the argument to cosine, and many people will, upon noticing one, completely block out the idea of looking for the other one.
 
Hurkyl said:
Well, this one can be tricky for some people -- there are two "obvious" ways to attack the problem via simplifying the argument to cosine, and many people will, upon noticing one, completely block out the idea of looking for the other one.

I know. After looking at Aequiveri's attempt, I was blocking the easy way for a while as well.
 

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