Consider the following homogeneous system of first order equations, dx/dt = ax + y, dy/dt = x − 2y, where a is a parameter. Suppose a = −2. Give the general solution of this system in the form x(t) = c1 u(t) r1 + c2 v(t) r2, where u, v, r1 and r2 should be found and c1, c2 are arbitrary constants. What is the nature of the critical point? Find the range of values of a for which the critical point is a saddle.

Consider the following homogeneous system of first order equations, dx/dt = ax + y, dy/dt = x − 2y, where a is a parameter. Suppose a = −2. Give the general solution of this system in the form x(t) = c1 u(t) r1 + c2 v(t) r2, where u, v, r1 and r2 should be found and c1, c2 are arbitrary constants. What is the nature of the critical point? Find the range of values of a for which the critical point is a saddle.

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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Question

Consider the following homogeneous system of first order equations,

dx/dt = ax + y, dy/dt = x − 2y, where a is a parameter.

Suppose a = −2. Give the general solution of this system in the form x(t) = c₁ u(t) r₁ + c₂ v(t) r_2,where u, v, r₁ and r₂ should be found and c₁, c₂ are arbitrary constants. What is the nature of the critical point? Find the range of values of a for which the critical point is a saddle.

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