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

[Knissp]

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.

 

[Mark44]

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.

 

[Knissp]

Cool thanks my solution worked.

 

[Mark44]

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.

 

[Knissp]

Yep the book's sol'n sure doesn't work. Thanks for the help! :)