Separable D.E., Nonlinear in terms of y(x) after integration

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Homework Help Overview

The discussion revolves around a separable differential equation that becomes nonlinear in terms of y after integration. Participants are exploring the implications of the equation ln(y) + y² = sin(x) + c₀.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various forms of expressing y in terms of x, including a suggestion involving the Lambert W function. There is also a question about the necessity of solving explicitly for y.

Discussion Status

The conversation is ongoing, with participants exploring different expressions and questioning the feasibility of obtaining an explicit solution for y. Some guidance has been offered regarding the use of the Lambert W function, but no consensus has been reached.

Contextual Notes

There is uncertainty regarding the requirement to solve for y explicitly, and participants are considering the implications of this in their approaches.

EtherealMonkey
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So, this is where I am stuck:

[tex]ln\left(y\right)+y^{2} = \sin{x}+c_{0}[/tex]

I am confrused... :blushing:
 
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Do you think that

[tex]y(x) =\pm\sqrt{2LambertW(2exp(2sin(x)+C)}/2[/tex]

is better?
 
Last edited by a moderator:
It's easier to express <x> in terms of <y>, wouldn't you say ?
 
EtherealMonkey, are you required to solve for y? That isn't always necessary or possible.
 
hmm, but is it possible to make an explicit form in terms of x?
 
kosovtsov just posted it...
 
waw, lambert, i need to study that thing
 

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