Ordrinary Differential Equation Problem

[Aequiveri]

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!

 

[Dick]

How about substituting u(x)=y(x)-x?

 

[Aequiveri]

Sorry, that was a lot simpler than I thought. Thanks for your help!

 

[Hurkyl]

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.

 

[Aequiveri]

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.

 

[Dick]

Sure. I think you were just overlooking the 'simple' before you launched into the 'hard' ways of solving it. Wrong order.

 

[Hurkyl]

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.

 

[Dick]

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.