anonymous
  • anonymous
Find the general solution of x'=(3, 2, -2, -2)x. (this is matrix, 3 and 2 on the left, -2 and -2 on the right.)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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Loser66
  • Loser66
\[\left[\begin{matrix}3&2\\-2&-2\end{matrix}\right]\] step 1: find eigenvalue: \[\left[\begin{matrix}3-\lambda&2\\-2&-2-\lambda\end{matrix}\right]\]characteristic equation: \[(3-\lambda)(-2-\lambda)+4=0\]
anonymous
  • anonymous
How do you find the eigenvalue?
Loser66
  • Loser66
hey, this information comes from linear algebra course. If you didn't take that course, how can you be in differential equation class?

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anonymous
  • anonymous
I suck at Linear Algebra. Sorry.
Loser66
  • Loser66
I didn't finish the first step yet.
anonymous
  • anonymous
Please continue.
Loser66
  • Loser66
so, do you know how to construct differential equation?
Loser66
  • Loser66
ok, from the equation above, I have the simplified form is \[\lambda^2-\lambda-2=0\\(\lambda -2)(\lambda+1)=0\\\lambda_1=2~~~and~~~ \lambda_2=-1 \] those are eigenvalues of the matrix. From those, plug into the second matrix, which I construct above to find eigenvectors
Loser66
  • Loser66
For \(\lambda_1=-1\) the matrix above (mine) becomes \[\left[\begin{matrix}4&2\\-2 &-1\end{matrix}\right]\], so, just calculate it by \[\left[\begin{matrix}4&2\\-2 &-1\end{matrix}\right] \left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)\]
Loser66
  • Loser66
so, -2a =b, if I chose a =1, so b = -2. therefore \[\left(\begin{matrix}1\\-2\end{matrix}\right)\]is eigenvector corresponds to eigenvalue -1 Can you imitate my stuff to find out the eigenvector corresponds to eigenvalue 2? We are near final answer, girl. PRACTICE, Please
anonymous
  • anonymous
How did you get -2a=b?
Loser66
  • Loser66
from the matrix |4 2 | | -2 -1| you can pick the last row time with vector (a,b) that mean -2a -b=0 or -2a =b
Loser66
  • Loser66
|dw:1378340651196:dw|
anonymous
  • anonymous
I got infinitely many solutions.
Loser66
  • Loser66
how? show me your work, please
anonymous
  • anonymous
You supposed to solve for a and b, right? I did elimination.
Loser66
  • Loser66
ok, then,?
Loser66
  • Loser66
oh, ya, I forgot, you are right
anonymous
  • anonymous
So I got 0=0.
Loser66
  • Loser66
you have infinite solution because eigenvector is a set of vectors whose spanned from the normalized one. That's why I choose a =1 to get b. We should choose the simplest one to calculate. You can choose whatever you want. and which any value of a, you have b. I'm so sorry, girl. I forgot explain you that part.
Loser66
  • Loser66
ok, continue finding eigenvector corresponds to eigenvalue 2. You do it. and show me your work. I have to check yours.
anonymous
  • anonymous
I got a+2b=0, -2a-4b=0, a=1, b=-1/2.
Loser66
  • Loser66
should avoid fraction, choose b =1 and a = -2, ok?
anonymous
  • anonymous
Okay, what's next?
Loser66
  • Loser66
then the general solution is x = |dw:1378341673374:dw|
anonymous
  • anonymous
But the answer in the textbook is c1*(1, 2)e^-t+c2*(2, 1)e^2t,
Loser66
  • Loser66
@SithsAndGiggles Please, check I don't know why she said that.
anonymous
  • anonymous
@Idealist, is this the same question as before? I think the error comes from the original matrix.
anonymous
  • anonymous
Yes.
Loser66
  • Loser66
oh, yeah, "3, 2 from the left" ha!! I waste your time,
anonymous
  • anonymous
@Loser66, the last time this was asked the matrix was \[\begin{bmatrix}3&-2\\2&-2\end{bmatrix}\] Right, @Idealist?
anonymous
  • anonymous
Yes.
Loser66
  • Loser66
@SithsAndGiggles thanks a tooon!! ha!!
anonymous
  • anonymous
Thanks. And why do you always call me stubborn?
Loser66
  • Loser66
no answer for that question
anonymous
  • anonymous
Why?
Loser66
  • Loser66
If you cannot handle my poke, just say " sister, don't call me that, I need a more beeeeaautiful nickname" hehehe, I may change my thought.
anonymous
  • anonymous
No, I can handle it.
Loser66
  • Loser66
ok, I have to study , cannot stay to chat with you. I spent a lot time for you, my younger sister.
Loser66
  • Loser66
I know you are willing to study, that's why I prefer you than other asker. Good luck, honey!!
anonymous
  • anonymous
I love you.
Loser66
  • Loser66
Never heard this sentence before \(\rightarrow\) don't know how to reply.

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