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Solution | an given x = 4.x = frany) y=-9 + S x² = g(ny).
22 4+ m2 (a) ana - Nulleling - given when NY=0 frany)=0 anol y- nullcling given when gmy)=0
-4 +5m? -0 =45m? ho yan Itsna 4+n? 4422 (0100) ar g(nay) as onco nullicling an direction of trajectories known as are given in table. f(and) >0 s nay)=0 / f (my) to glnasho 9(ny) 20 - genay) to
(6) fos finding equilibria point put f(ayl=0 grny)=0 and tolue them. y.n=0 = 4 and y t ro (8) put my in we get = *+5m2 = 0 = (x?s n+4)=0 =) = 1, 4.0 equilibrium points are (0.0), (11) and (4,4). as we already shown in the graph. (C) Ceneral, Jacobian at point P = (ny) (full) fy (f) I n J (1.91 - is given Il Now, P= (0,0) 569.
- . :) Real, oqual and an eigenvalug ar I and al, le nogestive in sign,
. (0,0) in asymptotically stable Nude. and s= (11) are real and on solving we get ergenually as 10+ 4 sio which of opponite sign. (1.1) in a saddle point.
.
10 = (4,4) J1 (44) = slona :) eigenvalue are lot auto they are Real, negatius and unequal : (4,4) is an anymptotically stable node. The graph in past (a) can be. y=542 yn
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