Differential Equations and Linear Algebra, 4th Edition

Section 7.6

1. 1. Repeated eigenvalue $\lambda =-3$, eigenvector $\mathbf{v}={\left[\begin{array}{rr}1& -1\end{array}\right]}^T;x_1(t)=(c_1+c_2+c_{2}t)e^{-3t}$, $x_2(t)=(-c_1-c_{2}t)e^{-3t}$

   

2. 2. Repeated eigenvalue $\lambda =2$, single eigenvector $\mathbf{v}={\left[\begin{array}{rr}1& 1\end{array}\right]}^T;$ $x_1(t)=(c_1+c_2+c_{2}t)e^{2t}\text{,}{x}_2(t)=(c_1+c_{2}t)e^{2t}$

   

3. 3. Repeated eigenvalue $\lambda =3$, eigenvector $\mathbf{v}={\left[\begin{array}{rr}-2& 2\end{array}\right]}^T;x_1(t)=(-2c_1+c_2-2c_{2}t)e^{3t}$, $x_2(t)=(2c_1+2c_{2}t)e^{3t}$

4. 4. Repeated eigenvalue $\lambda =4$, single eigenvector $\mathbf{v}={\left[\begin{array}{rr}-1& 1\end{array}\right]}^T;x_1(t)=(-c_1+c_2-c_{2}t)e^{4t}\text{,}{x}_2(t)=(c_1+c_{2}t)e^{4t}$

5. 5. Repeated eigenvalue $\lambda =5$, eigenvector $\mathbf{v}={\left[\begin{array}{rr}2& -4\end{array}\right]}^T;x_1(t)=(2c_1+c_2+2c_{2}t)e^{5t}$, $x_2(t)=(-4c_1-4c_{2}t)e^{5t}$

6. 6. Repeated eigenvalue $\lambda =5$, single eigenvector $\mathbf{v}={\left[\begin{array}{rr}-4& 4\end{array}\right]}^T$