Question

I need help with the defferential equation:
dy/dx = y/(y-x) , is it a linear D.E ?

Answer 1

http://tutorial.math.lamar.edu/Classes/DE/Definitions.aspx

then look at definition 11

Answer 2

tx but could you help me on this specefic problem?

Answer 3

can you write it in the form like definition 11 says?

Answer 4

I tried and I couldn't, so i don't think it's linear

Answer 5

ok so what do you think this one is? Homogenoud? exact? seprable?

Answer 6

It's not linear. Rewriting, you have
\[(y-x)~\frac{dy}{dx}=x\]
On the left, you have a function of the dependent variable multiplied by a power of its derivative.

While I'm still here, it's not separable: there's no way to right \(\dfrac{x}{y-x}\) as a product of functions \(f(x)g(y)\).

To determine whether it's homogeneous and/or exact, try writing it out in the form
\[M(x,y)~dx+N(x,y)~dy=0\]
If yes, it's homogeneous. Do you know how to determine whether an equation is exact?

Answer 7

the dm/dy should be equale to dn/dx...it is not exact eighter

Answer 8

you can use substitute to solve it
let y = Vx \[\frac{dy}{dx}=V +\frac{dV}{dx}V\]
so, \[V+\frac{dV}{dx}=\frac{Vx}{Vx-x}=\frac{V}{V-1}\]
\[\frac{dV}{dx}=\frac{V}{V-1}-V\]
solve for V and then plug the solution back to substitute above to solve for y.

Answer 9

\[\Large \frac{dy}{dx} =f(x,y)=f(tx,ty)\]
Your differential equation fits the condition to be linear with the substitution method, a prior mentioned by @Loser66 already.

Just consider to apply the initial test :-)

Answer 10

Thank you @Spacelimbus I take differential equation this summer, try to apply what I am taught. :)

Answer 11

\[\Huge \checkmark \]
Very good then (-: !