Question

Hi guys, I, again, need your help! Would someone make the equation ydx=[x+ sqrt(y^2-x^2)]dy into a homogeneous function! Thanks in advance!

Answer 1

I'd say that the presence of y^2 means non-linear.

Answer 2

Make the substitution x - y z, dx = z dy + y dz to convert to a variables-separable equation.

Answer 3

After the substitution is made, the result somplifies to
\[y dz =\sqrt{z ^{2}-1}dy.\]
Now you can separate the variables and integrate, then get it back to the original variables.

Answer 4

So, the equation that I gave is already a homogenous function?

Answer 5

Yes, it is. Of first degree.

Answer 6

Oh! Thanks abtrehearn! I thought the sqrt. part is not of the same degree!

Answer 7

Do,'t let the radical fool you. Make sure you understand how "homogeneous" is defined in this context.

Answer 8

abtrehearn, would you mind explaining how did the sqrt part be of first degree?

Answer 9

I would like to see the definition for homogeneous "in this context" (although I accept the change of variables approach).

Answer 10

Think of x and y as units of length, say meters. Then x^2 - y^2 is area, square meters. The square root of m^2 is m. This is the sort of thing done in dimensional analysis as a way to check answers for dimensional consistency.
A first-order differential equation is homogeneous if dy/dx can be expressed as a function of y/x only