432 Ordinary Differential EquationsE X ERCISES 10.2I n exercises 1 -8 calculate the eigenvalues of the given linear sys-
tem. Indicate whether the critical point is asymptotically stable,
neutrally stable, or unstable. Specify if the critical point is a node,
saddle point, spiral point, or center.- x ' = -2x + 3y 2. x ' = -x+2y
y' = - x + 2y y' = - 2x + 3y
3. x' = x - 2y 4. x ' = - x - 2y
y' = 2x - 3y y' = 5x + y
- x ' = -x+2y 6. x' = x - 2y
y' = -2x -y y' = 2x+ y
- x' = -5x -y + 2 8. x' = 3x-2y- 6
y' = 3x -y- (^3) y' = 4x-y + 2
In exercises 9- 15 find all real critical points. Compute the eigen-
values of the associated linear system at each critical point. Deter-
mine if the critical point is stable or unstable and, when possible,
specify its type (node, saddle point, or spiral point).
9. x' = -x + x y 10. x ' = x +y
y' = -y + xy y' = Y + x2
- x' = -x+xy 12. x' = x + 1
y' = y-2xy y' = 1-y 2
13. x ' = y2 - x2 1 4. x ' = y-xy
y' = 1 -x y' = y2 _ x2
- x ' = -x+xy