Invariant Lines: Differentialgleichungen

Question

Any help with it please

The following system is to be analyzed: xdot = y^3-4x ; ydot= y^3-y-3x

(a) Find all the fixed points and classify them.

(b) Show that the line x=y is invariant, meaning that any trajectory starting on this line stays on it subsequently.

(c) Show that |x(t)-y(t)|= 0 as t goes to infinity.

Answer 1

Betrachte direkt die Hilfsfunktion z=x-y und du hast z'=-z und fast schon alles

Gruß lul

Answer 2

Hallo,

(a) Find all the fixed points and classify them.

x' = y^3-4x

y'= y^3-y-3x

1. Linke Seite = 0 setzen:

1) 0 = y^3-4x

2) 0= y^3-y-3x

------------------------

1) 0 = y^3-4x -------->x= y^3/4

in 2 eingesetzt:

0= y^3-y-3x

0= y^3-y -(3/4) y^3 ------->0= (y^3/4) -y ------->

y₁= 0

y_2,3=± 2

-------->

Fixpunkte:

x₁= 0 ; y₁=0

x₂= 2  ; y₂= 2

x₃= -2 ; y₃= -2

2. Jacobi Matrix bilden

J(x,y)= \( \begin{pmatrix} -4 & 3 y^{2} \\ -3 &  3y^{2 } -1\end{pmatrix} \)

3. alle 3 Punkte in die Matrix einsetzen

4. Determinante = 0 setzen

---->Für P(0/0) habe ich erhalten -5/2 ±  i (√3)/2 ------>linke Halbebene ---->asymptotisch stabil

usw.