How can substitution be used to solve a differential equation?

[scrtajntman]

Homework Statement

The problem states: Use substitution to solve:

y'=1/(x+y)^2-1

Homework Equations

The Attempt at a Solution

I used the substitution of v=x+y

resulting in the answer y=[3(x-C)]^(1/3) - x

but I'm not too sure that's right

Can some help with the answer and the steps for getting it. It's just for a practice test so I'm not graded on it.

 

[rock.freak667]

Show all of the steps you took to get the correct answer. If you made an error in any step, we could point it out then.

 

[gabbagabbahey]

Homework Statement

The problem states: Use substitution to solve:

y'=1/(x+y)^2-1

Homework Equations

The Attempt at a Solution

I used the substitution of v=x+y

resulting in the answer y=[3(x-C)]^(1/3) - x

but I'm not too sure that's right

Can some help with the answer and the steps for getting it. It's just for a practice test so I'm not graded on it.

It looks like you got to $(x+y)^3=3x+C$ correctly (you probably had -3C instead of +C but it makes no difference since both are just undetermined constants) ? ...If so, this is how you should leave your answer (unless you are given a point on the curve y(x)). The reason being is that there are actually three roots to this cubic equation, and $y+x=\sqrt[3]{3x+C}$ is only one of them.

 

[scrtajntman]

Great! So overall I got the problem right. Thanks.