Question

Need help with this homogeneous equation
dy/dx= (x^2-y^2)/3xy.

Answer 1

put y=vx
\[\frac{ dy }{ dx }=v+x \frac{ dv }{ dx }\]

Answer 2

how did you get that? what did you substitute exactly?

Answer 3

http://tutorial.math.lamar.edu/Classes/DE/Substitutions.aspx
It is a common substitution.

Answer 4

have you tried that sub yet?

Answer 5

instead of in terms of y and x
you will have a differential equation in terms of v and x

Answer 6

I am trying something else out and I think I got it. The notation threw me off.

Answer 7

what did you try?

Answer 8

never mind the integral was really ugly
I tried multiplying by (y/x^3) to get in terms of (y/x)

Answer 9

ok well v=y/x can be written as vx=y (by multiply x on both sides)
wherever you see y put vx
but wherever you see dy/dx put v'x+v

Answer 10

to get from y=vx to y'=v'x+v you just need product rule

Answer 11

ok so we have dy/dx= (x^2-y^2)/3xy.

so replace dy/dx with y'=v'x+v and replacing y with vx we have
\[v'x+v=\frac{x^2-(vx)^2}{3x(vx)}\]

Answer 12

try to simplify that one fraction first

Answer 13

so I should get v'x+v=(x^2-(vx)^2)/3x(vx)

Answer 14

sweet got that

Answer 15

you should hopefully realize you have a separable differential equation

Answer 16

after of course simplifying that one fraction :)

Answer 17

When I simplified I got 1/3z - z/3
is that right? Want to make sure my algebra is correct

Answer 18

whoops 1/3v-v/3

Answer 19

\[\frac{x^2-(vx)^2}{3x(vx)}=\frac{x^2-v^2x^2}{3x^2v}=\frac{x^2(1-v^2)}{x^2(3v)} \\ =\frac{\cancel{x^2}(1-v^2)}{\cancel{x^2}(3v)}=\frac{1-v^2}{3v}\]
yep that looks like what you got
you just wrote it as two separate fractions

Answer 20

\[v'x+v=\frac{1-v^2}{3v}\]
\[\frac{dv}{dx}x+v=\frac{1-v^2}{3v} \\ \frac{dv}{dx} x =\frac{1-v^2}{3v}-v \]
\[\frac{dv}{dx}x=\frac{1-4v^2}{3v}\]
do you that we have a separable differential equation?

Answer 21

i will leave the separating to you but i will still be here if you have problems of course

Answer 22

awesome! Thank you so much! I should be able to do the separating. Its the simplifying I get stuck at :)

Answer 23

it is a homogeneous differential equation.
To solve it is best substitution.