ODE's: Find Change of Variables

[Leptos]

Homework Statement

xy' = yf(xy)

Homework Equations

The Attempt at a Solution

Attempt #1: F is a function of the product of x and y so I first thought of trying v = xy so dv = xdv + vdx but that would transform the equation into xy' = 1dv - vdx = yf(v).

Attempt #2: I tried vx = y and vy = x but nothing I've tried so far would transform the general form into something separable.

Attempt #3: I tried setting p = y' which would transform the original equation into xp = p²/2f(xy) since y = p²/2 which leads to 2x = pf(xy) but this is also a dead end.

I'm not sure how to deal with a function of the product x*y...

 

[SammyS]

Homework Statement

xy' = yf(xy)

The Attempt at a Solution

Attempt #1: F is a function of the product of x and y so I first thought of trying v = xy so dv = xdv + vdx but that would transform the equation into xy' = 1dv - vdx = yf(v).

I'm not sure how to deal with a function of the product x*y...

Regarding attempt #1:

If v=xy , then v' = y + xy' or equivalently, dv = ydx + xdy.

Thus, xy' = v' - y . Now replace y by v/x .

I don't know if this does much to solve the D.E., but it's lots different than what you had.

xv' = v(1 + f(v))

 

[Leptos]

Regarding attempt #1:

If v=xy , then v' = y + xy' or equivalently, dv = ydx + xdy.

Thus, xy' = v' - y . Now replace y by v/x .

I don't know if this does much to solve the D.E., but it's lots different than what you had.

xv' = v(1 + f(v))

Ah, it was a misuse/abuse of notation on my part then. Silly me.
Still, what's the thought process in the first place when we're dealing with a function of the product of two variables?

 

[SammyS]

This DE is now separable.

$$\frac{dv}{v(1+f(v))}=\frac{dx}{x}$$