Origin as the Only Critical Point: Solving Differential Equations

[Kamekui]

Homework Statement

Show the origin is the only critical point

Homework Equations

x'= -x-x³
y-= -y-y⁵

The Attempt at a Solution

I'm not really sure how to go about this. I missed a few lectures due to a medical issue, and now were at the end of the semester and I can't get in touch with the professor. Any help would be appreciated.

 

[DryRun]

Homework Equations

x'= -x-x³
y-= -y-y⁵

Your second equation doesn't make sense. The very first step when asking for help, is to always make sure that you have typed your question correctly. There is a minus after y. Is there supposed to be another constant or variable after the minus sign? Or is that minus sign not supposed to be there?

Anyway, to find the critical point, you should differentiate the function with respect to the independent variable and then equate to 0.

 

[LCKurtz]

Homework Statement

Show the origin is the only critical point

Homework Equations

x'= -x-x³
y-= -y-y⁵

The Attempt at a Solution

I'm not really sure how to go about this. I missed a few lectures due to a medical issue, and now were at the end of the semester and I can't get in touch with the professor. Any help would be appreciated.

Your second equation doesn't make sense. The very first step when asking for help, is to always make sure that you have typed your question correctly. There is a minus after y. Is there supposed to be another constant or variable after the minus sign? Or is that minus sign not supposed to be there?

Anyway, to find the critical point, you should differentiate the function with respect to the independent variable and then equate to 0.

It doesn't take much imagination to figure out that the second equation is $y'=-y-y^5$. This is a system of two differential equations. The stationary points are where $x'$ and $y'$ are simultaneously zero.

 

[DryRun]

It doesn't take much imagination to figure out that the second equation is $y'=-y-y^5$. This is a system of two differential equations. The stationary points are where $x'$ and $y'$ are simultaneously zero.

I kinda guessed, but i preferred to point it out to the OP. Previously, in other threads with typo errors, i had guessed and assumed, but it turned out that i was wrong, and i got flamed by the poster/s. So... I'm not doing the same mistake of assuming anything again.