### Author Topic: MT Problem 5  (Read 4145 times)

#### Victor Ivrii ##### MT Problem 5
« on: March 06, 2013, 09:10:42 PM »
Solve the system of ordinary differential equations
\begin{equation*}
\left\{
\begin{aligned}
&x'_t=5x-3y,\\
&y'_t=6x-4y.
\end{aligned}
\right.\end{equation*}

#### Matthew Cristoferi-Paolucci

• Jr. Member
•  • Posts: 10
• Karma: 8 ##### Re: MT Problem 5
« Reply #1 on: March 06, 2013, 09:57:57 PM »
Heres my solution

#### Jeong Yeon Yook

• Full Member
•   • Posts: 20
• Karma: 8 ##### Re: MT Problem 5
« Reply #2 on: March 06, 2013, 09:58:23 PM »
solution

#### Rudolf-Harri Oberg

• Jr. Member
•  • Posts: 9
• Karma: 9 ##### Re: MT Problem 5
« Reply #3 on: March 06, 2013, 10:03:41 PM »
We begin by finding eigenvalues for the systems matrix. We solve $(5-\lambda)(-4-\lambda)+18=\lambda^2-\lambda-2=0$. This yields $\lambda_1=2, \lambda_2=-1$. We now search for eigenvectors.

For $\lambda_1=2$, the eigenvector is $\xi_1=(1,1)$

For $\lambda_2=-1$, the eigenvector is $\xi_2=(1,2)$.

General solution for the system is $Y_G=c_1e^{2t}\xi_1+c_2e^{-t}\xi_2$
« Last Edit: March 06, 2013, 11:38:02 PM by Rudolf-Harri Oberg »

#### Devangi Vaghela

• Jr. Member
•  • Posts: 6
• Karma: 3 ##### Re: MT Problem 5
« Reply #4 on: March 06, 2013, 10:21:34 PM »
This is my solution