4.5 Exercises for Part 4

Section 4 exercises: Linear systems in 2 dimensions

• For the following differential equations, calculated the eigenvalues and eigenvectors of the eigenvalue equations, plot the phase portraits and classify the systems. (Answers )

• $\dot{x}=y,\dot{y}=-2x-3y $

• $\dot{x}=5x+10y,\dot{y}=-x-y $

• $\dot{x}=3x-4y,\dot{y}=x-y $

• $\dot{x}=-3x+2y,\dot{y}=x-2y $

• $\dot{x}=5x+2y,\dot{y}=-17x-5y $

• $\dot{x}=-3x+4y,\dot{y}=-2x+3y $

• $\dot{x}=4x-3y,\dot{y}=8x-6y $

• $\dot{x}=y,\dot{y}=-x-2y $

• What will Thando and Marsha’s future be in the following situations?

• $\dot{T}=M,\dot{M}=-T+M$

• $\dot{T}=a M,\dot{M}=b T \left(\text{ for any } \text{ real } a \text{ and } b\right)$

• $\dot{T}=a T+b M,\dot{M}=-b T-a M $

• $\dot{T}=a T+b M,\dot{M}=b T+a M . What \text{ if } a=b?$

• $\dot{T}=0,\dot{M}=a T+b M $