Solving First Order Diff Eq: y^2 = x^2 + cx^3?

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Homework Help Overview

The discussion revolves around solving a first-order differential equation given by dy/dx = (2x + y - 1)^2. The original poster attempts to derive a solution and compares it with a solution provided in a textbook, which is y^2 = x^2 + cx^3.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts a substitution method to simplify the differential equation and derives a solution. They question the validity of their solution in comparison to the textbook answer, suggesting a potential typo or error in the book.

Discussion Status

Some participants have confirmed that the original poster's solution satisfies the differential equation, indicating that it may be correct. There is a suggestion to check the textbook's solution, with acknowledgment that different forms of solutions can exist in differential equations.

Contextual Notes

Participants note that solutions to differential equations can sometimes appear different yet still be valid, as they may differ by a constant. There is an implication that the textbook's solution may not align with the derived solution, but no definitive resolution is reached.

Knissp
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Homework Statement



dy/dx = (2x + y - 1)^2


Homework Equations





The Attempt at a Solution



Let u = 2x + y
du = 2 + dy/dx
dy/dx = du/dx - 2

dy/dx = (2x + y - 1)^2
so du/dx - 2 = (u-1)^2
du/dx = (u-1)^2 + 2
du / ((u-1)^2 + 2) = dx

1/sqrt(2) * arctan ((u-1)/sqrt(2)) = x + c

1/sqrt(2) * arctan ((2x + y -1)/sqrt(2)) = x + c

y = sqrt(2) * tan(sqrt(2) * x + C) + 1 - 2x

BUT the answer in the back of the textbook is y^2 = x^2 + cx^3. Did I mess up or is it a typo? Thank you.
 
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You have two answers. Check them to see if they satisfy dy/dx = (2x + y - 1)^2. If you find that your answer satisfies this DE, that's pretty good evidence that the book's answer is wrong. I don't see anything obviously wrong with your work.
 
Cool thanks my solution worked.
 
If you feel really ambitious, you could check the book's solution. Sometimes with differential equations it's possible to get what look like completely different solutions, but they both work. The key is that they differ by a constant.

As an example, sin^2(x) and -cos^2(x) look to be very different, but differ only by a constant.
 
Yep the book's sol'n sure doesn't work. Thanks for the help! :)
 

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