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.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...

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