# How Can You Transform and Solve This System of Equations?

• MHB
• karush
In summary, the given system can be rewritten as $x_1''+ x_1'+8x_1= 0$ and the solutions for $x_1$ and $x_2$ with initial conditions are $x_1=\dfrac{11}{3}e^{2t}-\dfrac{2}{3}e^{-t}$ and $x_2=\dfrac{11}{6}e^{2t}-\dfrac{4}{3}e^{-t}$ respectively. The graph of the solution in the $x_1x_2$-plane for $t > 0$ can be sketched on Desmos.
karush
Gold Member
MHB
ck for typos
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

(a)$\quad x_1''-x_1'-2x_1=0$
(b)$\quad x_1=\dfrac{11}{3}e^{2t}-\dfrac{2}{3}e^{-t},\quad x_2=\dfrac{11}{6}e^{2t}-\dfrac{4}{3}e^{-t}$

You are given that $x_1'= 3x_1- 2x_2$ so $x_1''= 3x_1'- 2x_2'$.
You are also given that $x_2'= 2x_1- 2x_2$ so $x_1''= 3x_1'- 2(2x_1- 2x_2)= 3x_1'- 4x_1+ 4x_2$.

From $x_1'= 3x_1- 2x_2$, $2x_2= 3x_1- x_1'$ so
$x_1''= 3x_1'- 4x_1+ 4(3x_1- x_1')= -x_1'- 8x_1$

$x_1''+ x_1'+8x_1= 0$.

Country Boy said:
You are given that $x_1'= 3x_1- 2x_2$ so $x_1''= 3x_1'- 2x_2'$.
You are also given that $x_2'= 2x_1- 2x_2$ so $x_1''= 3x_1'- 2(2x_1- 2x_2)= 3x_1'- 4x_1+ 4x_2$.

From $x_1'= 3x_1- 2x_2$, $2x_2= 3x_1- x_1'$ so
$x_1''= 3x_1'- 4x_1+ 4(3x_1- x_1')= -x_1'- 8x_1$

$x_1''+ x_1'+8x_1= 0$.

ok but the book answer $x_1''-x_1'-2x_1=0$

so if we rewrite the eq with $x=e^{γt}$ we have

$\left(\left(e^{γt}\right)\right)''\:-\left(\left(e^{γt}\right)\right)'\:-2e^{γt}=0$

not sure what we do with IV

Country Boy said:
You are given that $x_1'= 3x_1- 2x_2$ so $x_1''= 3x_1'- 2x_2'$.
You are also given that $x_2'= 2x_1- 2x_2$ so $x_1''= 3x_1'- 2(2x_1- 2x_2)= 3x_1'- 4x_1+ 4x_2$.

From $x_1'= 3x_1- 2x_2$, $2x_2= 3x_1- x_1'$ so
x_1''+ x_1'+8x_1= 0$. This is an error. Since$2x_2= 3x_1- x_1'$,$4x_2= 6x_1- 2x_1'\$

ok I see the substitution

## 1. What is an IVP with a system of equations?

An IVP (initial value problem) with a system of equations is a mathematical problem that involves finding a solution to a set of equations, given a set of initial conditions. The initial conditions specify the values of the variables at a specific point in time or space.

## 2. What is the purpose of solving an IVP with a system of equations?

The purpose of solving an IVP with a system of equations is to find a mathematical model that accurately describes a real-world phenomenon or system. This model can then be used to make predictions or analyze the behavior of the system.

## 3. What are the steps to solve an IVP with a system of equations?

The general steps to solve an IVP with a system of equations are:

1. Write down the system of equations.
2. Determine the initial conditions.
3. Find the general solution to the system of equations.
4. Use the initial conditions to find the specific solution.

## 4. What are some common techniques used to solve an IVP with a system of equations?

Some common techniques used to solve an IVP with a system of equations include separation of variables, substitution, and elimination. Other techniques such as Laplace transforms and numerical methods may also be used depending on the complexity of the problem.

## 5. How can I check if my solution to an IVP with a system of equations is correct?

To check if your solution to an IVP with a system of equations is correct, you can substitute the solution into the original equations and verify that it satisfies all of the equations. Additionally, you can also use graphing or numerical methods to compare your solution to the behavior of the system in real-life situations.

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