Question

Determine an appropriate substitution and solve xdy/dx = y Ln(xy)

Answer 1

Is this a homogeneous function if so can I use the substitution y=ux.
Can I then use the method of seperable equations.

Answer 2

You sure the ques is right? It seems quite complex to me.

Answer 3

yeah the question is correct I re wrote it though as dy/dx= (y/x)*ln(xy)
the ln(xy) = lnx + ln y
the three methods in class we pretty much have used are solving homogeneous differential equation. solving a bernoulli DE and reduction to separation of variables which I don't understand. My friend for this problem I think used a U substitution.

Answer 4

The answer is also y(x) = e^(c1(x-1))/x

Answer 5

after one substitution of vy =v, i got this in the form of variable separation. did you try this substitution ?

Answer 6

* xy=v

Answer 7

I was going to try that but I get confused about what happens does the dy/dx change then into dv/dx and then what happens to the (y/x)(lnv)

Answer 8

xy = v
differentiate both sides, what you get ?

Answer 9

x dy + y dx = v dx or v'

Answer 10

wouldn't it be , xdy+ydx = dv

Answer 11

Yeah I believe so the problems in the book just never give good examples. so then you get something like dv=xdy+ydx(y/x)(ln (v)) I am just confused on how the xdy+ydx cancels out with the (y/x) do you get 1?

Answer 12

from xdy+ydx = dv you can find x dy/dx =.... ?
y ln (xy) changes to (v/x)ln v

Answer 13

xdy+ydx = dv

x dy/dx + y = dv/dx
so, x dy/dx = dv/dx - y = dv/dx -v/x

Answer 14

dv/dx -v/x= (v/x)ln v
this is variable separable, if you try it.

Answer 15

Thanks alot u have been a real help I hope some day I will understand differential equations better. My teacher is just horrible and we just go over bad examples in the book and usually have to spend hours watching youtube to find out simple methods. I think I get how to set up this equation now.

Answer 16

Welcome ^_^
with some practice, this DE will be piece of cake :)
oh and by the way,
\(\huge \color{red}{\text{Welcome to Open Study}}\ddot\smile\)

Answer 17

Goku saves the day yet again! \m/