Variation of Parameters on a system of Differential Eqs (Simple question)

[Saladsamurai]

Homework Statement

Okay so when solving a system of D.E.s using Variation of Parameters I know that first I find the complementary solution Xc and then do a bunch a of crap after that using the fundamental matrix.

Now I just came across a problem with repeated roots, so I just want to clarify that I am correct in saying that if the complementary solution looks like this:

$$X_c=c_1\left(\begin{array}{c}1\\1\end{array}\right)+c_2[\left(\begin{array}{c}1\\1\end{array}\right)t+\left(\begin{array}{c}1\\0\end{array}\right)]$$Then the fundamental matrix looks like this:

$$\Phi(t)=\left(\begin{array}{cc}1 & t+1\\ 1 & t\end{array}\right)$$
Just a yes or no will do... (if it's no, I am in trouble!)

Thanks!

 

[Saladsamurai]

Anyone? I just want to make sure before I go using this...

 

[Saladsamurai]

Well I thought it was a simple question...

 

[Saladsamurai]

Tina... eat the food!

 

[Saladsamurai]

eat the food!

 

[HallsofIvy]

Homework Statement

Okay so when solving a system of D.E.s using Variation of Parameters I know that first I find the complementary solution Xc and then do a bunch a of crap after that using the fundamental matrix.

Now I just came across a problem with repeated roots, so I just want to clarify that I am correct in saying that if the complementary solution looks like this:

$$X_c=c_1\left(\begin{array}{c}1\\1\end{array}\right)+c_2[\left(\begin{array}{c}1\\1\end{array}\right)t+\left(\begin{array}{c}1\\0\end{array}\right)]$$


Then the fundamental matrix looks like this:

$$\Phi(t)=\left(\begin{array}{cc}1 & t+1\\ 1 & t\end{array}\right)$$



Just a yes or no will do... (if it's no, I am in trouble!)

Thanks!

Does it help you to point out that

$$\left(\begin{array}{cc}1 & t+1\\ 1 & t\end{array}\right)\left(\begin{array}{c}c_1 \\ c_2\end{array}\right)= \left(\begin{array}{c}c_1+ c_2(t+1) \\ c_1+ c_2t\end{array}\right)$$