dx/dt = ax + by
dy/dt = bx + cy
1. First compute the eigenvalues for this system as functions of a, b, c.

2. Now consider the case b = 0 . Compute the eigenvalues for this special case and determine the exact a,c-values where this system has different types of behavior, i.e., spiral sinks, sources, saddles, etc. Give the formulas for the regions in which the system has these characteristics. Then draw an accurate picture of the a,c-plane indicating the regions (i.e., the points (a,c)) where the two parameter family

dx/dt = ax
dy/dt = cy

has the corresponding behavior.

Question

dx/dt = ax + by
dy/dt = bx + cy
1. First compute the eigenvalues for this system as functions of a, b, c.

2. Now consider the case b = 0 . Compute the eigenvalues for this special case and determine the exact a,c-values where this system has different types of behavior, i.e., spiral sinks, sources, saddles, etc. Give the formulas for the regions in which the system has these characteristics. Then draw an accurate picture of the a,c-plane indicating the regions (i.e., the points (a,c)) where the two parameter family

dx/dt = ax
dy/dt = cy

has the corresponding behavior.