Solving y' + 4xy - y² = 4x² - 7 with substitution v = y/x

[FeynmanIsMyHero]

Hi,

I'm having trouble with this ODE:

y' + 4xy - y^2 = 4x^2 - 7

=> y' = 4x^2 - 4xy + y^2 - 7
= (2x - y)^2 - 7

=> x(dv/dx) + v = (2x-vx)^2 - 7
= x^2(2-v)^2 - 7


I assume this ODE is of the homogeneous type, so I've substituted v=y/x and gotten thus far, but, what's the next step?

 

[ehild]

Hi,

I'm having trouble with this ODE:

y' + 4xy - y^2 = 4x^2 - 7

=> y' = 4x^2 - 4xy + y^2 - 7
= (2x - y)^2 - 7

Try to replace 2x-y by v and solve for v. .

ehild

 

[Benny]

Hmm this looks exactly like one of the questions on my assignment. Don't know of you are from University of Melbourne but if you are then you are ignoring the explicit instruction that we're supposed to write up the solutions to the assignment independently.

In any case if you are from the same uni as I am then a hint that I can give you w/o actually telling you how to do the question is to go back to the problem sheets. There is a question which requires the same technique.

You are on the right track btw.

 

[FeynmanIsMyHero]

Thanks, I've figured it out now!