Integrating seperable equation

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Homework Help Overview

The discussion revolves around two separable differential equations: dy/dx = y * e^(sinx + cos(y)) and dy/dx = sin(x^y). Participants are exploring the integration of these equations and the nature of their separability.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the separation of variables for the first equation and express difficulty in integrating e^(sinx). There is an attempt to use substitution and integration by parts. For the second equation, some participants question its separability and clarify that it does not fit the criteria.

Discussion Status

The discussion is active, with participants offering insights into the challenges of integrating the first equation and clarifying the nature of the second equation. There is acknowledgment that the first equation leads to non-elementary integrals, and some participants suggest alternative approaches while others express uncertainty.

Contextual Notes

Participants note that the first equation may not yield elementary solutions and that the second equation is not separable, which raises questions about the methods applicable to each problem.

astonmartin
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Homework Statement



Separable equations

dy/dx = y * e^(sinx +cosy)

and

dy/dx = sin(x^y)

The Attempt at a Solution



For the first problem, I did dy/dx = y * e^(sinx) * e^(cosy) and separated. However, I can't figure out how to integrate e^(sinx)dx on the right. Did I do somehting wrong?

I have no idea what to do on the second one
Thanks for the help!
 
Last edited:
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Help pleeeease
 
For the first one, my first impression would have been to let u= sin x, giving [tex]\int \frac{e^u}{\sqrt{1-u^2}} du[/tex] and then tried an integration by parts. Checking with the Integrator online however, it seems there is no elementary antiderivative for that function, so don't be surprised if the Integration by parts doesn't work out. Still try it though, because the Integrator has been wrong before.

The Second one isn't actually separable.
 
The second, y'= sin(x^y), is NOT separable.

The first is separable but gives integrals that cannot be integrated as elementary functions.
[tex]\int \frac{dy}{ye^{cos(y)}}= \int e^{sin(x)} dx[/tex]
is the best you can do.

Thanks, Gib Z.
 
Last edited by a moderator:
Small correction - The exponential term on the LHS should be in the denominator. =]
 

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