Reduce y'' + y''' Sin(y) = 0 to first order and solve

[Ethan1990]

y''+y'''Sin(y)=0

I have been stuck on this question for a few days now, I think i need to use substitution such as y'=w and y''= w(dw/dy) however I m just not making sense of this and cant find resources online that are helping. Can anyone please help me solve this equation.

 

[stapel]

y''+y'''Sin(y)=0

I have been stuck on this question for a few days now, I think i need to use substitution such as y'=w and y''= w(dw/dy) however I m just not making sense of this and cant find resources online that are helping. Can anyone please help me solve this equation.

Please reply showing all your steps in at least one of your efforts from the last few days, so folks can see what you're trying and where things are going sideways.

Please be complete. Thank you!

 

[Ethan1990]

Well if I replaced y''= w. I think the equation will become w + w (dw/dy)Sin(y) = 0. Then using some algebra I can turn it into w (dw/dy)Sin(y) = -w
Then to (dw/dy)Sin(y) = -w/w then to (dw/dy)Sin(y) = -1
After this Im not to sure what to do. I know that this is a third-order nonlinear equation with the independent variable missing I just can't find any information anywhere on how to solve these. All the information I can find involves second order and usually with some starting conditions like y(0)=e^2x or something like that. I'm not looking to be told the answer, maybe even the working to a similar question so I have something to go off. Thank you

 

[HallsofIvy]

Well if I replaced y''= w. I think the equation will become w + w (dw/dy)Sin(y) = 0. Then using some algebra I can turn it into w (dw/dy)Sin(y) = -w
Then to (dw/dy)Sin(y) = -w/w then to (dw/dy)Sin(y) = -1

What's wrong with dw/dy= -1/sin(y) and then dw= -dy/sin(y).

Recalling that the derivative of cos(y) is negative sin(y), I would multiply both top and bottom of the right side by sin(y) an write the equation as dw= (-sin(y)/sin^2(y))dy= (-sin(y)/(1- cos^2(y))dy. Now let u= cos(y) so this becomes
dw= du/(1- u^2)

After this Im not to sure what to do. I know that this is a third-order nonlinear equation with the independent variable missing I just can't find any information anywhere on how to solve these.

The whole point was to reduce it to a first order differential equation and you have done that! Don't worry about "third order" any more!

All the information I can find involves second order and usually with some starting conditions like y(0)=e^2x or something like that. I'm not looking to be told the answer, maybe even the working to a similar question so I have something to go off. Thank you

 

[nant]

Perhaps in this case it is possible to use the Newton's Method (here such an example) for solving nonlinear equations? This method is based on the idea of linearization and it is a generalization of Newton's Method of solving nonlinear equations. This example is a good training for your knowledge of mathematics.