# Find the approximate linear ODE system

## Main Question or Discussion Point

dx/dt = x-y^2 dy/dt= x^2 -xy -2x
For each critical point, find the approximate linear OD system that is valid in a small neighborhood of it.

I found the critical points which are (0,0),(4,2),(4,-2) but have no idea how to do the above question! please help!

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S.G. Janssens
Just calculate the Jacobian matrices of the vector field at the three critical points. Your approximate linear ODE system at the critical point $(x_c,y_c)$. is going to be of the form $\dot{u}(t) = A(x_c,y_c)u(t)$ where $A(x_c,y_c)$ is the Jacobian matrix.

To check that this system is valid in a small neighborhood, verify that $A(x_c,y_c)$ has no spectrum on the imaginary axis. Which theorem do you use here?

• NiallBucks
I'm not sure about the theorem

S.G. Janssens
I'm not sure about the theorem
It's the Hartman–Grobman theorem. You should consider looking it up, it's worthwhile.

HallsofIvy
Homework Helper
To "linearize" an equation simply means to replace any non-linear function by a linear approximation. But the only linear approximations to $x^2$ and $xy$ are "0". At (0, 0), dx/dt= x- y^2 linearizes to dx/dt= x and dy/dt= x^2- xy- 2x to dy/dt= -2x.

About (4, 2), let u= x- 4 and v= y- 2 so that x= u+ 4, y= v+ 2, dx/dt= du/dt, and dy/dt= dv/dt. The equations become du/dt= u+4- (v+2)^2= u+ 4- v^2- 4v- 4 which linearizes to du/dt= u- 4v and dv/dt= (u+ 4)^2- (u+ 4)(v+ 2)- 2(u+ 4)= u^2+ 8u+ 16- uv- 4v- 2u- 8- 2u- 8= u^2- uv+ 4u- 6v- uv which linearizes to dv/dt= 4u- 6v.

• NiallBucks
S.G. Janssens
To "linearize" an equation simply means to replace any non-linear function by a linear approximation.
Why (and when, and when not) the linearisation says something meaningful about the original system in a neighborhood of the critical point, is less simple, though.

Mark44
Mentor
dx/dt = x-y^2 dy/dt= x^2 -xy -2x
For each critical point, find the approximate linear OD system that is valid in a small neighborhood of it.

I found the critical points which are (0,0),(4,2),(4,-2) but have no idea how to do the above question! please help!
(4, -2) is NOT a critical point.

Also, you posted a question in another forum section about the critical points of this system. Your other question was posted in the right section (Calculus & Beyond under Homework & Coursework Questions). Please take care to post homework questions there, not here in the technical math sections.

Also, don't post essentially the same question in multiple forum sections.