Non homogeneous linear system

In summary: To solve x^5- 3x^4+ 4x^2- 3x+ 1= 0, you would need to find all the solutions to x^5- 3x^4+ 4x^2- 3x+ 1= 0 and then use the fact that those solutions are all linear combinations of x^4,x^3,x^2,x.
  • #1
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Homework Statement


Verify that the given vector is the general solution of the corresponding homogeneous system and then solve the nonhomogenous system. Assume that t>0.

tx' =
|2 -1|x + |1- t^2|
|3 -2| |2t |

General solution:
x =
c1| 1|t + c2| 1|t^-1
| 1| |3|

This won't show up correctly but the first eigen vector is (1,1) and the second is (1,3)

Homework Equations





The Attempt at a Solution


So I have been solving non homogeneous linear systems all night with no problems however when I got to this problem I got stumped because there is a t in front of x'. I solved for the eigen values and eigen vectors and the eigen vectors match what they have but I have the general solution by solving it without acknowledging the t in front of x'. I thought that maybe I could just divide my general solution I got by t and then I would have the same answer, however that doesn't work because the solution provided by the problem doesn't have any e^t's in it. So I am not quite sure on what is going on exactly.
 
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  • #2
hi simmonj7! :smile:

(use the CODE tag for matrices … it's not what it's there for, but it does work! :wink:)

for the homogeous solution, i suspect you're meant to assume a solution of the form tn

for the particular solution, if it wasn't a matrix, you'd try a polynomial P(t) …

so try a vector of two polynomials, (P(t),Q(t)) :wink:
 
  • #3
The problem is
[tex]t\begin{bmatrix}x \\ y\end{bmatrix}'= \begin{bmatrix}2 & -1 \\ 3 & -2\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}+ \begin{bmatrix} 1- t^2 \\ 2t\end{bmatrix}[/tex]

and you want to show that
[tex]c_1\begin{bmatrix}1 \\ 1\end{bmatrix}t+ c_2\begin{bmatrix}1 \\ 3\end{bmatrix}t^{-1}[/tex]
is the general solution.

Both you and tiny-tim talk about "solving" the system but you are not asked to solve it! You are only asked to verify that the given solution works and that is a far easier problem.
(Which would be easier, to solve [itex]x^5- 3x^4+ 4x^2- 3x+ 1= 0[/itex] or to show that x= 1 is a solution?)

To show that the given function is the general solution, you need to use the fact that the general solution of a linear equation is a general linear combination of two independent solutions.

So you need to show three things:
1) That
[tex]\begin{bmatrix}1 \\ 1\end{bmatrix}t= \begin{bmatrix}t \\ t\end{bmatrix}[/tex]
satisfies the equation by putting it and its derivative into the equation.

2) That
[tex]\begin{bmatrix}1 \\ 3\end{bmatrix}t^{-1}= \begin{bmatrix}t^{-1} \\ 3t^{-1}\end{bmatrix}[/tex]
satisfies the equation by putting it and its derivative into the equation.

3) That those two functions are independent (that one is not a multiple of the other).
 

1. What is a non homogeneous linear system?

A non homogeneous linear system is a system of equations in which the constant term is not equal to zero. This means that the equations do not form a homogeneous system and will have a unique solution.

2. How is a non homogeneous linear system different from a homogeneous linear system?

In a homogeneous linear system, the constant term is equal to zero, which means that all the equations are equal to zero and have infinitely many solutions. However, in a non homogeneous linear system, the constant term is not equal to zero and has a unique solution.

3. What are some methods for solving a non homogeneous linear system?

There are several methods for solving a non homogeneous linear system, including substitution, elimination, and matrix methods such as Gaussian elimination and Cramer's rule. These methods involve manipulating the equations to isolate the variables and solve for their values.

4. Can a non homogeneous linear system have no solution?

Yes, a non homogeneous linear system can have no solution if the equations are inconsistent, meaning they contradict each other and have no common solution. This can happen if the constant terms are not compatible with each other.

5. What are some real-world applications of non homogeneous linear systems?

Non homogeneous linear systems have many real-world applications, such as in economics, physics, and engineering. They can be used to model systems with changing conditions or external influences, such as population growth, chemical reactions, and circuit networks.

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