Solving Differential Equation: Substitution Method

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Homework Help Overview

The discussion revolves around solving a differential equation using the substitution method. Participants are examining the steps taken by the original poster and exploring the implications of their approach.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts substitution to simplify the differential equation but encounters difficulties in solving for the variables. Some participants question the necessity of finding an explicit solution, suggesting that an implicit solution might suffice. Others assert that an explicit solution is achievable.

Discussion Status

The discussion is ongoing, with participants providing different perspectives on the validity of the original poster's method. There is a suggestion that the integration steps may have been correct, leading to a potentially acceptable solution, though no consensus has been reached.

Contextual Notes

Participants are considering the implications of implicit versus explicit solutions in the context of differential equations, and there is an acknowledgment of possible arithmetic mistakes in the original poster's calculations.

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Homework Statement


Solve: http://img384.imageshack.us/img384/60/clipboard01az7.jpg

Homework Equations


DE stuff

The Attempt at a Solution


I started by doing substitution: x=U+3/2 and y=W-1/2

So, it gave me http://img522.imageshack.us/img522/3172/clipboard02ly8.jpg ,[/URL] but http://img167.imageshack.us/img167/8297/clipboard03db9.jpg ,[/URL] so http://img175.imageshack.us/img175/3223/clipboard04xt2.jpg .[/URL]

Therefore, http://img237.imageshack.us/img237/2264/clipboard05iu8.jpg

And, http://img377.imageshack.us/img377/615/clipboard06ez2.jpg .[/URL] Substitution z=W/U. Giving, http://img139.imageshack.us/img139/5639/clipboard07hb7.jpg (*).

And, http://img399.imageshack.us/img399/2373/clipboard08qj2.jpg

Rearranging gives http://img114.imageshack.us/img114/1830/clipboard09tm2.jpg .[/URL] Substitute in (*)

http://img140.imageshack.us/img140/8541/clipboard10bz2.jpg => http://img140.imageshack.us/img140/6706/clipboard11ao0.jpg => http://img135.imageshack.us/img135/8163/clipboard12ee3.jpg

Dividing 1/(1+z2) and -z/(1+z2) and integrating gives:

http://img521.imageshack.us/img521/3638/clipboard13nj5.jpg .[/URL] But now I'm stuck :( because I can't solve this for z and go back to y and x :( maybe I made an arithmetical mistake or maybe there is a better method?
Thank you
 
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I don't know how correct the method you detailed is but a lot of times for a differential equation, an implicit solution is sufficient.
 
Surely there is a way to solve it explicitely
 
Why "surely". You don't need to solve
arctan z-\frac{1}{2}ln(z^2+ 1)= ln(U)+ C

z= W/U so that is
arctan W/U-\frac{1}{2}ln(\frac{W^2}{U^2}+ 1)= ln(U)+ C
and U= x- 3/2, W= y+ 1/2 so
arctan\frac{y+ 1/2}{x-3/2}- \frac{1}{2}ln(\frac{(y+1/2)^2}{(x-3/2)^2}= ln(x- 3/2)+ C
That's a perfectly good solution. (Assuming your integration was correct.)