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Classify fixed points non homogeneous system of linear differential equations

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Homework Statement



[tex]\dot{x}=2x+5y+1, \dot{y}=-x+3y-4[/tex]

Homework Equations




The Attempt at a Solution


Well, if system was: [tex]\dot{x}=2x+5y, \dot{y}=-x+3y[/tex] we let a=2, b=5, c=-1, d=3.
Then p = a + d and q =ad - bc and we investigate [tex] p^2-4q[/tex]
Don't know what to do when [tex]\dot{x}=2x+5y+1, \dot{y}=-x+3y-4[/tex]
Thanks
 
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Answers and Replies

  • #2
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Why don't you try making a change in coordinates?

Solve the system of equations:
[tex] 2x + 5y = -1; -x + 3y = 4;[/tex]
for x and y. Assume we obtain the solutions [itex]x=x_0, y=y_0[/itex]. Then a substitution of the form [itex]x^* = x + x_0; y^* = y + y_0[/itex] will give you a set of homogeneous ODEs in a shifted coordinate system. Fixed points of this new set of ODEs are related to the old set by the coordinate transformations above.
 
  • #3
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good intelligent answer thank you
 

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