6.3E: Exercises for Section 6.3

Exercise \(\PageIndex{2}\)

a. Find the general solution of \( x'_1=2x_1, x'_2=3x_2\) using the eigenvalue method (first write the system in the form \(\vec{x}'=A\vec{x}\) ).

b. Solve the system by solving each equation separately and verify you get the same general solution.

Exercise \(\PageIndex{3}\)

Find the general solution of \(x'_1 = 3x_1 +x_2, x'_2 =2x_1+4x_2\) using the eigenvalue method.

Exercise \(\PageIndex{4}\)

Find the general solution of \(x'_1 =x_1-2x_2, x'_2=2x_1 + x_2\) using the eigenvalue method. Do not use complex exponentials in your solution.

Exercise \(\PageIndex{5}\)

a. Compute eigenvalues and eigenvectors of \(A= \left[ \begin{array}{ccc} 9 & -2 & -6 \\ -8 & 3 & 6 \\ 10 & -2 & -6 \end{array} \right] \).

b. Find the general solution of \(\vec{x}'=A\vec{x}\).

Exercise \(\PageIndex{6}\)

Compute eigenvalues and eigenvectors of \( \left[ \begin{array}{ccc} -2 & -1 & -1 \\ 3 & 2 & 1 \\ -3 & -1 & 0 \end{array} \right].\)

Exercise \(\PageIndex{7}\)

Let \(a, b, c, d, e, f\) be numbers. Find the eigenvalues of \( \left[ \begin{array}{ccc} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{array} \right].\)

Exercise \(\PageIndex{8}\)

a. Compute eigenvalues and eigenvectors of \(A= \left[ \begin{array}{ccc} 1 & 0 & 3 \\ -1 & 0 & 1 \\ 2 & 0 & 2 \end{array} \right] \).

b. Solve the system \(\vec{x}'=A\vec{x}\).

Answer

a. Eigenvalues: \(4,\:0,\:-1\) Eigenvectors: \(\left[\begin{array}{c}{1}\\{0}\\{1}\end{array}\right]\), \(\left[\begin{array}{c}{0}\\{1}\\{0}\end{array}\right]\), \(\left[\begin{array}{c}{3}\\{5}\\{-2}\end{array}\right]\)

b. \(\vec{x}=C_{1}\left[\begin{array}{c}{1}\\{0}\\{1}\end{array}\right] e^{4t}+C_{2}\left[\begin{array}{c}{0}\\{1}\\{0}\end{array}\right]+C_{3}\left[\begin{array}{c}{3}\\{5}\\{-2}\end{array}\right]\)

Exercise \(\PageIndex{9}\)

a. Compute eigenvalues and eigenvectors of \(A= \left[ \begin{array}{cc} 1 & 1 \\ -1 & 0 \end{array} \right].\)

b. Solve the system \(\vec{x}'=A\vec{x}\).

Answer

a. Eigenvalues: \(\frac{1+\sqrt{3}i}{2}\), \(\frac{1-\sqrt{3}i}{2}\), Eigenvectors: \(\left[\begin{array}{c}{-2}\\{1-\sqrt{3}i}\end{array}\right]\), \(\left[\begin{array}{c}{-2}\\{1+\sqrt{3}i}\end{array}\right]\)

b. \(\vec{x}=C_{1}e^{t/2}\left[\begin{array}{c}{-2\cos\left(\frac{\sqrt{3}t}{2}\right)}\\{\cos\left(\frac{\sqrt{3}t}{2}\right)+\sqrt{3}\sin\left(\frac{\sqrt{3}t}{2}\right)}\end{array}\right]+C_{2}e^{t/2}\left[\begin{array}{c}{-2\sin\left(\frac{\sqrt{3}t}{2}\right)}\\{\sin\left(\frac{\sqrt{3}t}{2}\right)-\sqrt{3}\cos\left(\frac{\sqrt{3}t}{2}\right)}\end{array}\right]\)

Exercise \(\PageIndex{10}\)

Solve \( x'_1=x_2, x'_2=x_1\) using the eigenvalue method.

Answer

\(\vec{x}=C_{1}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{t}+C_{2}\left[\begin{array}{c}{1}\\{-1}\end{array}\right]e^{-t}\)

Exercise \(\PageIndex{11}\)

Solve \( x'_1=x_2, x'_2=-x_1\) using the eigenvalue method.

Answer

\(\vec{x}=C_{1}\left[\begin{array}{c}{\cos(t)}\\{-\sin(t)}\end{array}\right]+C_{2}\left[\begin{array}{c}{\sin(t)}\\{\cos(t)}\end{array}\right]\)