Question

Please show work!!

(1 point) Take the system x' = 4x – xy, y = 5y + x2. How many critical points are there? What is the critical point with the largest x-coordinate? ( The linearization at this point is | tion at this point is [*] =A (Where A is where A is ul A = At this point the behavior is For behavior write one of "saddle", "source", "sink", "spiral sink", "spiral source", "center".

Answer 1

. : 4-ay - y' = 59 +32 G We know that Suppose Praound in an Rolated conical point of the systom. - fray) ay = gay) and f and a digte at P. then linearization of the systom at p. is the linen System dy = fou + fy (P) duga (P)2 + Jy CPU Now from ean and dor = 45-ay (4-9)X = 0 x=0, 4-4 dly = 54+02 54+0²=0 when go, yoo Y=4 then sx4 +0²=0 there are (0,0), ( 2555, 4) 2²=- 20 and (-2) 55,4) to je x20 C x=+) 255 flogy)= 40-ay 9 (0,4)= Syra? fox (ay)=4-9 . galay) = 20 fy (0,4)= -x gylony) = 5 at (oo) for 10,0) = 4 ga(0,0)=0 fy (0,0)=0 94 (0,0) = 5 at (815,4) fx (@its, 4) = 4-4=0 galajjs, 4) = 455 fy (915s, 4) = -gj gylairs, 4) = 5

Od (-2355,9) for (-ajj, 1) = 4 450 fy(-a55s, 4) = aus 3:03,) - Ox(-0355) - -4355 gyland) - 5 Jy (155, 1) = 5 Now matrix room at cool 14 01 Here da 4,5 Eigen value (Real Positive unequal Cimpropor nodal. Source) Matrix toom at Coits ,t) B o - 2015 2250-4000=0 Luis 5 Matrix foom at (-2555,4) -To 2155 x²-54-4020 -yis 5 the linearization at this point is. JPN -:) **) TEJN 9j15) Blosso) at (+2jJ5,4) so 22.50-40=0 X= 5+25+160 5+5185 (Real, unequal opposite Sign) Saddle point)