Solving y'=y-sin(2x): Integral of -sin(2x)e^x Stuck, Help!

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Homework Help Overview

The discussion revolves around solving the differential equation y' = y - sin(2x) using an integrating factor. Participants are exploring the integration of the term -sin(2x)e^x as part of their solution process.

Discussion Character

  • Mathematical reasoning, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • One participant identifies the integrating factor as e^x and attempts to compute the integral of -sin(2x)e^x using integration by parts. Another participant points out the presence of the same integral on both sides of the equation, suggesting a direct solution. There is also a mention of potential oversight regarding the integrating factor and boundary conditions.

Discussion Status

The discussion is active, with participants providing insights and clarifications. Some guidance has been offered regarding the relationship between the integrals, and there is acknowledgment of potential errors in the initial setup. Multiple interpretations of the problem are being explored, but no consensus has been reached.

Contextual Notes

Participants are navigating assumptions about the integrating factor and the boundaries of the differential equation, which may affect their approaches and conclusions.

koolrizi
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I found the integrating factor to be e^x for DE y'=y-sin(2x)
Now i am stuck at integral of -sin(2x)e^x
Can you help me with it? I tried using integration by parts and i get to following
integral(sin(2x)(e^x)=sin(2x)e^x-2cos(2x)e^x-4*integral(sin(2x)(e^x)

I am stuck. Help! Thanks
 
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Notice that you have the expression "integral(sin(2x)(e^x))" on both sides of your equation. You can therefore solve for it directly, and be done.
 
directly? I think i am not seeing something
 
The answer to this DE is 1/5(sin(2x)+2cos(2x) and i am not seeing how to get to this from what i have
 
OH ok i got it...wasnt seeing that lol...THanks
 
also careful with your results. looks like you missed a couple small details. your integrating factor is actually $e^{-x}$ and you should have the boundaries of the LHS of your equation(unless they're supposed to be =0)
 

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