Question

how to determine if this diff eq is homogeneous (or not)
2lntdt-ln(4y^2)dy=0

Answer 1

all right let's try this:
2*ln(t) - ln(4y^2) * (dy/dt) = 0
2*ln(t) = ln(4y^2)*(dy/dt)
dy/dt = 2*ln(t)/ln(4y^2)

Answer 2

let dy/dt = f(t,y)
The function will be homogenous if:
f(xt,xy) = f(t,y)

Answer 3

As far as i can tell it's not homogenous, because you have y^2 and that will leave u with an x^2 on the bottom and only x on top

Answer 4

the way i learned, it's homogeneous if x^n*M(t,y) and x^n*N(t,y) share the same power n. so substituting (t,y) with (xt,xy) for each term separately, M(t,y)dt becomes M(xt,xy)dt. pull out the x, and it's x^2*M(t,y) but i can't pull the x out so how do i prove they are not homogeneous? i understand what you're saying, how do i write it out mathematically?