Integrating Initial Condition Problems: Finding the Constant of Integration

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Homework Help Overview

The discussion revolves around solving the differential equation y' = (xysinx)/(y+1) with the initial condition y(0) = 1. Participants are exploring the integration process and the implications of the initial condition on finding the constant of integration.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the separation of variables and integration steps. There is a focus on the resulting equation y + ln(y) = sinx - xcosx + C and the challenges in isolating y. Questions arise regarding the ability to determine the constant of integration given the initial condition.

Discussion Status

Some participants have provided guidance on substituting the initial condition into the equation to evaluate the constant. Multiple interpretations of the integration process and the handling of the initial condition are being explored, but there is no explicit consensus on the method to isolate y or the implications of the constant.

Contextual Notes

Participants note the complexity of expressing y explicitly in the derived equation and question whether the initial condition allows for a unique determination of the constant of integration.

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Homework Statement



They give me y ' = (xysinx)/ (y+1) , y(0) = 1

Homework Equations



So I just separated and integrated

The Attempt at a Solution



I'm 99 percent sure I'm OK up till here. I just wanted to get an explanation for something.

I was wondering my answer is y + ln(y) = sinx -xcosx +C
So there is no way to write this so I just have y on the right. So what am I to say? There is an example similar in my book they does this mean I can't find the constant?
 
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Jbreezy said:

Homework Statement



They give me y ' = (xysinx)/ (y+1) , y(0) = 1

Homework Equations



So I just separated and integrated

The Attempt at a Solution



I'm 99 percent sure I'm OK up till here. I just wanted to get an explanation for something.

Sure. This differential equation is variable separable.

I was wondering my answer is y + ln(y) = sinx -xcosx +C
So there is no way to write this so I just have y on the right. So what am I to say? There is an example similar in my book they does this mean I can't find the constant?

I think that the initial condition is, at x=0, y=1. Well the equation can be simplified as,

ln(yey) = sinx -xcosx +C
yey = ke(sinx -xcosx), where k is another constant. But does this lead to anything ?
 
ln(y(e^y)) = sinx -xcosx +C
How did you get this?
 
Jbreezy said:

Homework Statement



They give me y ' = (xysinx)/ (y+1) , y(0) = 1

Homework Equations



So I just separated and integrated

The Attempt at a Solution



I'm 99 percent sure I'm OK up till here. I just wanted to get an explanation for something.

I was wondering my answer is y + ln(y) = sinx -xcosx +C
So there is no way to write this so I just have y on the right. So what am I to say? There is an example similar in my book they does this mean I can't find the constant?

Just put x=0 and y=1 in that last equation to evaluate C.
 
This last equation ?
y + ln(y) = sinx -xcosx +C
 
Yes.
 
Got it thanks
 

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