- #1

karush

Gold Member

MHB

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https://photos.app.goo.gl/eRfYNAVK1jnBgSCu8

https://photos.app.goo.gl/8C9sJ9UgZbxXgP4P9

Boyce Book

(a) Transform the given system into a single equation of second order.

(b) Find $x_1$ and $x_2$ that also satisfy the given initial conditions.

(c) Sketch the graph of the solution in the $x+1x_2$-plane for $t > 0$.

$\begin{array}{rrr}

x_1'=3x_1-2x_2 & x_1(0)=3\\

x_2'=2x_1-2x_2 & x_2(0)=\dfrac{1}{2}

\end{array}$

ok this is not a homework assignment but I reviewing before taking the class

also not sure if desmos can plot the answer

if there appears to be a typo go to the links above

the book seemed a little sparce on a good example to work with so...there was an exaple on page 362 but I couldn't follow it

well one way is to first rewrite x' to $x'=Ax$ where

$A=\left[\begin{array}{rrr}

3&-2\\

2&-2

\end{array}\right]$

so far

book answer

(a)$\quad x_1''-x_1'-2x_1=0$

(b)$\quad x_1=\dfrac{11}{3}e^{2t}-\dfrac{2}{3}e^{-t},\quad x_2=\dfrac{11}{6}e^{2t}-\dfrac{4}{3}e^{-t}$