Finding equilibrium for differential equations x ′ = x 2 + y 2 − 5 y ′ = x 2 + 2 y 2 − 9My goal is to find the equilibrium points. ( 5 − y 2 , ± 2 ), ( ± 9 11 , 5 − x 2 ) , ( 9 − 2 y 2 , ± 2 ) and ( ± 11 3 , 9 − x 2 2 )

Question

Finding equilibrium for differential equations      x    ′    =      x    2    +      y    2    −  5      y    ′    =      x    2    +  2      y    2    −  9My goal is to find the equilibrium points.  (      5    −          y      2        ,  ±  2  ),       (    ±                            9          11                      ,          5      −              x        2              )  ,   (      9    −    2          y      2        ,  ±  2  ) and       (    ±                            11          3                      ,                            9          −                      x            2                          2              )

vikafa4g · Accepted Answer

This is a linear system of ODEs, so you can treat them as a system of equations. In this case, the system is simple enough that we do not need to use linear algebra. We can simply solve (1)       x    2    +      y    2    =  5 and (2)       x    2    +  2      y    2    =  9Note that   x  =      −    1    ,    1   and   y  =      −    2    ,    2   solve (1) and you can plug them into (2) to verify, so your steady-state solutions are       p    1    =  (  −  1  ,  −  2  )  ,      p    2    =  (  −  1  ,  2  )  ,      p    3    =  (  1  ,  −  2  )  ,      p    4    =  (  1  ,  2  )

Isaiah Owens · Accepted Answer

You have      x    2    +      y    2    =  5      x    2    +  2      y    2    =  9Subtract one from the other and you get   y  =  ±  2. Plug that back in, and you get   (  ±  1  ,  ±  2  ) to be your steady states.

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