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(a) Consider a system of two first-order ordinary differential equations: ₁ =e-2y + sin(a² y2 - 1), y2 = tanh(-y₁ + y2) Linearise the system of ODE close to the (y₁, y2) = (0, 0) equilibrium. For a = 1 the phase portrait of the linearised system is: (5 marks) Ostable node OUnstable node Osaddle OUnstable focus with spiral out OCentre OStable focus with spiral in (b) For which real value of a the system of ODES y₁ = ln(1 - y₁) + a²y2/(1- y₁), y2 = cos(y₁)y₁ + y + ln(1 + y2) when linearised, displays a centre around the equilibria (y1, y2) = (0, 0)? (5 marks) OAny value of a Oa > 1 Oa > 0 Oa < 1

(a) Consider a system of two first-order ordinary differential equations: ₁ =e-2y + sin(a² y2 - 1), y2 = tanh(-y₁ + y2) Linearise the system of ODE close to the (y₁, y2) = (0, 0) equilibrium. For a = 1 the phase portrait of the linearised system is: (5 marks) Ostable node OUnstable node Osaddle OUnstable focus with spiral out OCentre OStable focus with spiral in (b) For which real value of a the system of ODES y₁ = ln(1 - y₁) + a²y2/(1- y₁), y2 = cos(y₁)y₁ + y + ln(1 + y2) when linearised, displays a centre around the equilibria (y1, y2) = (0, 0)? (5 marks) OAny value of a Oa > 1 Oa > 0 Oa < 1

Advanced Engineering Mathematics
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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(a) Consider a system of two first-order ordinary differential equations: ₁ = e-² + sin(a² y2 - 1), y2 = tanh(-y₁ + y2) Linearise the system of ODE close to the (y1, y2) = (0,0) equilibrium. For a = 1 the phase portrait of the linearised system is: (5 marks) OUnstable focus with spiral out OCentre OStable focus with spiral in Ostable node OUnstable node Osaddle (b) For which real value of a the system of ODES y₁ = ln(1 - y₁) + a²y2/(1-y₁), (y1, y2) = (0, 0)? (5 marks) OAny value of a Oa > 1 Oa> 0 Oa < 1 y2 = cos(y₁)y₁ + y² + In(1 + y₂) when linearised, displays a centre around the equilibria
Transcribed Image Text:(a) Consider a system of two first-order ordinary differential equations: ₁ = e-² + sin(a² y2 - 1), y2 = tanh(-y₁ + y2) Linearise the system of ODE close to the (y1, y2) = (0,0) equilibrium. For a = 1 the phase portrait of the linearised system is: (5 marks) OUnstable focus with spiral out OCentre OStable focus with spiral in Ostable node OUnstable node Osaddle (b) For which real value of a the system of ODES y₁ = ln(1 - y₁) + a²y2/(1-y₁), (y1, y2) = (0, 0)? (5 marks) OAny value of a Oa > 1 Oa> 0 Oa < 1 y2 = cos(y₁)y₁ + y² + In(1 + y₂) when linearised, displays a centre around the equilibria
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