Given the system below find the fundamental solution
The answer should be:
$$x_1 = e^t \bigl( \begin{smallmatrix} 1 \\ -1 \end{smallmatrix} \bigr) ; x_2 = te^t \bigl( \begin{smallmatrix} 1 \\ -1 \end{smallmatrix} \bigr) + e^t \bigl( \begin{smallmatrix} 1 \\ 0 \end{smallmatrix} \bigr)$$
However, I do not understand where the last term for $x_2$ comes from. I found the eigenvalues and eigenvectors of the matrix given by the system and simple got that:
$$x_1 = e^t \bigl( \begin{smallmatrix} 1 \\ -1 \end{smallmatrix} \bigr) ; x_2 = te^t \bigl( \begin{smallmatrix} 1 \\ -1 \end{smallmatrix} \bigr)$$
There aren't any functions dependent on some other variable like $t$ so I didn't think that the equation: $$X' = AX + F(t)$$ would be used, and I don't understand how it would be used in this case. Am I missing something? Or is some other equation used for this problem ?
Thank you!
The given solution is indeed correct, and in order to understand this you need to know how to construct a second linear independent solution in the case of a double root of the characteristic polynomial. Your textbook will probably have a section on this, and otherwise you can always have a look at Paul's online notes. http://tutorial.math.lamar.edu/Classes/DE/RepeatedEigenvalues.aspx