Math Challenges

Submissions for Problem #86

a) Find the eigenvalues and eigenvectors.
b) Find the general solution.

\[ \frac{d\mathbf{Y}}{\differentialD t}=\begin{pmatrix}2 & 4\\ 3 & 6\end{pmatrix}\mathbf{Y} \]

zapwai

Solution: \( \displaylines{\left(2-\lambda\right)\left(6-\lambda\right)-12=0\\ \lambda^2-8\lambda=\left(\lambda-8\right)\lambda=0\\ \lambda=0,8\\ v_1=\begin{pmatrix}-2\\ 1\end{pmatrix},v_2=\begin{pmatrix}2\\ 3\end{pmatrix}\\ \mathbf{Y}=c_1\begin{pmatrix}-2\\ 1\end{pmatrix}+c_2e^{8t}\begin{pmatrix}2\\ 3\end{pmatrix}} \)

alston3a1

Solution: \( eigen1=0eigen2=8 \)