The suitable Liapunov function for the systema) V(x, y) = x² + y²b) V(x, y) = x² + 2y²c) V(x, y) = 2x² + y²d) V (x,y) = 1⁄2 x² + y²dxdt=−x³ + xy²,dt= -2x²y - y³ is

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(x2y-)水 + (2x +y+) dy~ 0 let メーuー1, dx = du yov-l, dy ody LINE 1 %3D LINE 2 (4-1+ 2y - 2-1) du t (2u-2+ v -I+1) dy =0 (u+ 2x) du + (2u + v) dy レINE 3 LINE 4 ミO let u= 2y du こ 2dy + yd2 レINE S LINE 6 (ZY + 2Y)(zdvt ydz) + (22y +y)dx =0 LINE 8 (2+2)( zdv + vdz) + (22 t1) dy = 0 LINE 7 ○=Xp(人+人そ2 LINE 9 (22+ 2え)d +(22+1)d+ (2+2) vd2= 0 Qコ入や(1+ぞて +P^(て+そ) + Pモ (て十そ) LINE 10 %3D (2²+ 27 t 1) av + (2+2)ydz =0 2+2-dz 22+42 + In (x) + Y2 In (22t 42+1) = C LIANE 14 In (vZ) + In (2²+ 4z+) =C In Lv (22 +42 +1]-'C LINE II LINE 12 LINE 13 LINE IS LINE Ib ye (z-+42+)こC v2 (Wん + 4u +)=C u2 + 4uy t yZ LINE 7 LINE 18 LINE 19 v( 42 + 4uY+x2=C (メ)2+4[xtり(リー)+ (yー)2-C レINE 20 LINE 21 %3D
Check if the system with the following input-output relation are i) Memory/memory lessii) Causal /non-causala) y(t) =x^4(t) + x(t-3) b) y(n) = x(n-2) +3y(n-1)c) y(t) = A cos (2ㅠft + x(t))
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Classify (if possible) each critical point of the nonlinear system: x' = xy - 3y - 4 y' = xy - y
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3 2 2) Z³ - 3x ²y + 6 xyz = 0; use the method of partial differentiation 1 to find ax да да ду I
9. Let y=3x+2 a) typically we write points in this scenario as (x,y), not (y,x), why is that? b) on a horizontal axis, represent 3 things: x=-2, x=4, and Δx=4-(-2). c) on a vertical axis, represent 3 things: y=-4, y=14, and Δy=14-(-4). d) draw a sketch of y=f(x) which includes the points (-2,-4) and (4,14). Mark and label Δx and Δy for those two points on this graph. e) calculate that rate, then write the change equation for x and y f) if the change in x is 2, what is the change in y? g) if the change in y is -12, what is the change in x? h) if we swapped x and y (with x on the vertical axis so x=g(y)), what would the rate be?
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The suitable Liapunov function for the system a) V(x,y) = x² + y² b) V (x,y) = x² + 2y² c) V (x, y) = 2x² + y² d) V(x,y) = 1⁄2 x² + y² dt +xy², dt = -2x²y-y³ is