- #1
WendysRules
- 37
- 3
Homework Statement
Find all solutions of the given differential equations: ## \frac{dx}{dt} =
\begin{bmatrix}
6 & -3 \\
2 & 1
\end{bmatrix} x ##
Homework Equations
The Attempt at a Solution
So, we just take the determinate of A-I##\lambda## and set it equal to 0 to get the eigenvalues of 3, and 4. Then, we plug our eigenvalues one at a time into A-I##\lambda## and solve it to equal the 0 vector.
For 3, I get ##
\begin{bmatrix}
3 & -3 \\
2 & -2
\end{bmatrix} ## Which implies that ##x_1 = x_2## so I can just pick
##\begin{bmatrix}
1 \\
1
\end{bmatrix} ## to be my vector? This is the part I am confused about, I'm not sure that's what I actually do! Do I just say for ##\lambda## = 3, my solution is ## e^{3t} \begin{bmatrix}
1 \\
1
\end{bmatrix} ##
Similarly for ##\lambda = 4 ## i find my matrix to be
##
\begin{bmatrix}
2 & -3 \\
2 & -3
\end{bmatrix} ## which i see that ##2x_1=3x_2## this, i can just pick
##\begin{bmatrix}
2 \\
3
\end{bmatrix} ## Thus, my solution for ##\lambda = 4## is just ##e^{4t}
\begin{bmatrix}
2 \\
3
\end{bmatrix} ##
Thanks for all the help, I'm just unsure this is the method I should be going about to solve these types of problems! (Also, if anyone knows of an algebraic calculator in which I could check my solutions for these types of problems, I would also appreciate that)