determine a region in the xy plane...

• Jan 23rd 2011, 05:33 PM
slapmaxwell1
determine a region in the xy plane...
ok so im supposed to determine a region in the xy plane that would have a unique solution whose graph passes through a point in the region. (x+3y)y' = 2x - y

im not sure where to start this problem. am i supposed to find a point that satisfies the equation? any help would be appreciated.

thanks
• Jan 24th 2011, 12:17 AM
SENTINEL4
Maybe you should solve the differential equation at the beginning??
• Jan 24th 2011, 01:03 AM
FernandoRevilla
Quote:

Originally Posted by slapmaxwell1
ok so im supposed to determine a region in the xy plane that would have a unique solution whose graph passes through a point in the region. (x+3y)y' = 2x - y

Consider the domain:

$D=\left\{{(x,y)\in \mathbb{R}^2:x+3y\neq 0}\right\}$

and

$y'=f(x,y)=\dfrac{2x-y}{x+3y}$

We have:

$(i)\quad f:D\rightarrow{\mathbb{R}}$ is continuous.

$(ii)\quad \dfrac{{\partial f}}{{\partial y}}\in\mathcal{C}^1(D)$

Now, use a well known existence and uniqueness theorem.

Fernando Revilla
• Jan 25th 2011, 11:02 PM
slapmaxwell1
thanks fernando. i went back and redid the problem. and yes i looked at where the domain would not exist and used that to help me to find my domain.