Question

Let u' = Au be the linear approximation of the following system of non-\linear differential equation x' = F(x,y); y' = G(x, y). Let $\lambda_1, \lambda_2$ be two eigenvalues of the 2 × 2 matrix A at the critical point $(x_0, y_0)$. Then A. $(x_0, y_0)$ is an improper node if $\lambda_1, \lambda_2$ are real and have same sign. B. $(x_0, y_0)$ is a spiral point if $\lambda_1, \lambda_2$ are complex numbers. C. $(x_0, y_0)$ is an proper/improper node if $\lambda_1, \lambda_2$ are real and equal D. $(x_0, y_0)$ is a saddle point if $\lambda_1, \lambda_2$ are real and unequal. E. $(x_0, y_0)$ is unstable if $\lambda_1, \lambda_2$ are purely imaginary

          Let u' = Au be the linear approximation of the following system of non-\linear differential equation
x' = F(x,y); y' = G(x, y).
Let $\lambda_1, \lambda_2$ be two eigenvalues of the 2 × 2 matrix A at the critical point
$(x_0, y_0)$. Then
A. $(x_0, y_0)$ is an improper node if $\lambda_1, \lambda_2$ are real and have same sign.
B. $(x_0, y_0)$ is a spiral point if $\lambda_1, \lambda_2$ are complex numbers.
C. $(x_0, y_0)$ is an proper/improper node if $\lambda_1, \lambda_2$ are real and equal
D. $(x_0, y_0)$ is a saddle point if $\lambda_1, \lambda_2$ are real and unequal.
E. $(x_0, y_0)$ is unstable if $\lambda_1, \lambda_2$ are purely imaginary

Let u' = Au be the linear approximation of the following system of non-differential equation
x' = F(x,y); y' = G(x, y).
Let λ1, λ2 be two eigenvalues of the 2 × 2 matrix A at the critical point
(x0, y0). Then
A. (x0, y0) is an improper node if λ1, λ2 are real and have same sign.
B. (x0, y0) is a spiral point if λ1, λ2 are complex numbers.
C. (x0, y0) is an proper/improper node if λ1, λ2 are real and equal
D. (x0, y0) is a saddle point if λ1, λ2 are real and unequal.
E. (x0, y0) is unstable if λ1, λ2 are purely imaginary

Added by Jose Francisco H.

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