
# 10.6E: Constant Coefficient Homogeneous Systems III (Exercises)



are the real and imaginary parts of

$e^{\alpha t}(\cos\beta t+i\sin\beta t)({\bf u}+i{\bf v}).\nonumber$

27. Show that if the vectors $${\bf u}$$ and $${\bf v}$$ are not both $${\bf 0}$$ and $$\beta\ne0$$ then the vector functions

$\bf y_1=e^{\alpha t}({\bf u}\cos\beta t-{\bf v}\sin\beta t)\quad \mbox{ and }\quad \bf y_2=e^{\alpha t}({\bf u}\sin\beta t+{\bf v}\cos\beta t)\nonumber$

are linearly independent on every interval.

28. Suppose $${\bf u}=\left[\begin{array}{c}{u_{1}}\\{u_{2}}\end{array}\right]$$ and $${\bf v}=\left[\begin{array}{c}{v_{1}}\\{v_{2}}\end{array}\right]$$ are not orthogonal; that is, $$({\bf u},{\bf v})\ne0$$.

1. Show that the quadratic equation $({\bf u},{\bf v})k^2+(\|{\bf v}\|^2-\|{\bf u}\|^2)k-({\bf u},{\bf v})=0\nonumber$ has a positive root $$k_1$$ and a negative root $$k_2=-1/k_1$$.
2. Let $${\bf u}_1^{(1)}={\bf u}-k_1{\bf v}$$, $${\bf v}_1^{(1)}={\bf v}+k_1{\bf u}$$, $${\bf u}_1^{(2)}={\bf u}-k_2{\bf v}$$, and $${\bf v}_1^{(2)}={\bf v}+k_2{\bf u}$$, so that $$({\bf u}_1^{(1)},{\bf v}_1^{(1)}) =({\bf u}_1^{(2)},{\bf v}_1^{(2)})=0$$, from the discussion given above. Show that ${\bf u}_1^{(2)}={{\bf v}_1^{(1)}\over k_1} \quad \text{and} \quad {\bf v}_1^{(2)}=-{{\bf u}_1^{(1)}\over k_1}.\nonumber$
3. Let $${\bf U}_1$$, $${\bf V}_1$$, $${\bf U}_2$$, and $${\bf V}_2$$ be unit vectors in the directions of $${\bf u}_1^{(1)}$$, $${\bf v}_1^{(1)}$$, $${\bf u}_1^{(2)}$$, and $${\bf v}_1^{(2)}$$, respectively. Conclude from (a) that $${\bf U}_2={\bf V}_1$$ and $${\bf V}_2=-{\bf U}_1$$, and that therefore the counterclockwise angles from $${\bf U}_1$$ to $${\bf V}_1$$ and from $${\bf U}_2$$ to $${\bf V}_2$$ are both $$\pi/2$$ or both $$-\pi/2$$.

## Q10.6.4

In Exercises 10.6.29-10.6.32 find vectors $${\bf U}$$ and $${\bf V}$$ parallel to the axes of symmetry of the trajectories, and plot some typical trajectories.

29. $${\bf y}'=\left[\begin{array}{cc}{3}&{-5}\\{5}&{-3}\end{array}\right]{\bf y}$$

30. $${\bf y}'=\left[\begin{array}{cc}{-15}&{10}\\{-25}&{15}\end{array}\right]{\bf y}$$

31. $${\bf y}'=\left[\begin{array}{cc}{-4}&{8}\\{-4}&{4}\end{array}\right]{\bf y}$$

32. $${\bf y}'=\left[\begin{array}{cc}{-3}&{-15}\\{3}&{3}\end{array}\right]{\bf y}$$

## Q10.6.5

In Exercises 10.6.33-10.6.40 find vectors $${\bf U}$$ and $${\bf V}$$ parallel to the axes of symmetry of the shadow trajectories, and plot a typical trajectory.

33. $${\bf y}'=\left[\begin{array}{cc}{-5}&{6}\\{-12}&{7}\end{array}\right]{\bf y}$$

34. $${\bf y}'=\left[\begin{array}{cc}{5}&{-12}\\{6}&{-7}\end{array}\right]{\bf y}$$

35. $${\bf y}'=\left[\begin{array}{cc}{4}&{-5}\\{9}&{-2}\end{array}\right]{\bf y}$$

36. $${\bf y}'=\left[\begin{array}{cc}{-4}&{9}\\{-5}&{2}\end{array}\right]{\bf y}$$

37. $${\bf y}'=\left[\begin{array}{cc}{-1}&{10}\\{-10}&{-1}\end{array}\right]{\bf y}$$

38. $${\bf y}'=\left[\begin{array}{cc}{-1}&{-5}\\{20}&{-1}\end{array}\right]{\bf y}$$

39. $${\bf y}'=\left[\begin{array}{cc}{-7}&{10}\\{-10}&{9}\end{array}\right]{\bf y}$$

40. $${\bf y}'=\left[\begin{array}{cc}{-7}&{6}\\{-12}&{5}\end{array}\right]{\bf y}$$