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Consider the system of differential equations

dx/dt= x+ y=1
dy/dt =1- x^2 +y^2

Required:
Sketch the x-nullcline, where solutions must travel vertically. Identify the regions in the plane where solutions will move toward the right, and where solutions move toward the right.

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Answer:

Step-by-step explanation:

Given that:

the differential equations:

[tex]\dfrac{dx}{dt}= x+y = 1 \\ \\ \dfrac{dy}{dt}= 1-x^2+y^2[/tex]

For x-nullcline;

[tex]\implies \dfrac{dx}{dt} =0 \\ \\ \implies x+y-1[/tex]

From the image attached below, the sketch of the x-nullcline was carefully drawn and the regions were identified.

So, x-increases at the time when [tex]\dfrac{dx}{dt}>0[/tex]

[tex]\implies x+y -1 >0[/tex]

Thus, the solution move towards the right for  x+y>1

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