Can homogeneous substitution solve this differential equation?

[jdawg]

Homework Statement

x(dy/dx) - y = sqrt(xy +x²)

Homework Equations

The Attempt at a Solution

I got up to this point: u=y/x

dy/dx = (sqrt(xy+x²))/x + y/x

And then the solution shows this:

dy/dx = y/x + (y/x+1)^½

Please help, I don't understand how they got to that point.

 

[jdawg]

Also, sorry about the bad formatting. Where did they move the square root symbol and fraction symbol?

 

[epenguin]

It's fairly elementary algebra that was probably dinned into you at some point and you have forgotten.

√(xy + x²)/x = √(xy + x²)/√((x²) = √[(xy + x²)/x²] = ...

You might last time have lost sight of the why, just drilled to do that: hope something comes back but this time instead look to see how it makes sense.

 

[jdawg]

Thank you so much! Haha its been a minute since I've had algebra. Great explanation! :D