General solution to system of equations and initial value problem

• paperweight11
In summary, the given system of equations in matrix form with initial condition x(0) = <2, -1> can be solved using the eigenvalues and eigenvectors method. The eigenvalues are 1 and 3, and the corresponding eigenvectors are <1, 0> and <1, 1>.
paperweight11

Homework Statement

Find the general solution for the following systems of equations, a solution to the
initial value problem and plot the phase portrait.
--> this is in matrix formx' =

1 2
0 3

all multiplied by x.

also, x(0) =

2
-1

Homework Equations

Determinant, etc.

The Attempt at a Solution

I first use the eigen values and get it to:

eigenvalue1 = 1; eigenvalue2= 3

however, when I start with 1, my matrix is pretty screwed up:

0 2
0 2

So when I'm trying to figure out the values for the eigenvector I am confused. I don't know what to put it as.

Any help would be appreciated. Thanks.

hi paperweight11!
paperweight11 said:
I first use the eigen values and get it to:

eigenvalue1 = 1; eigenvalue2= 3

how did you get those?

(those would be the eigenvalues for 1,0,,0,3)

They are also the eigenvalues for
$$\begin{bmatrix} 1 & 2 \\ 0 & 3\end{bmatrix}$$
since the characteristic equation is just
$$\left|\begin{array}{cc}1- \lambda & 2 \\ 0 & 3- \lambda\end{array}\right|= (1- \lambda)(3- \lambda)= 0$$
(The eigenvalues for any triangular matrix are just the values on the diagonal.)

The eigenvectors must satisfy
$$\begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}x \\ y\end{bmatrix}$$
so x+ 2y= x and y= y. x+ 2y= x reduces to y= 0 which gives <1, 0> as an eigenvector, and
$$\begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}3x \\ 3y\end{bmatrix}$$
so x+ 2y= 3x and 3y= 3y so that y= x which gives <1, 1> as an eigenvector.

( Your
0 2
0 2
is the same as 2y= 0 so the eigenvector is <1, 0>.)

1. What is a general solution to a system of equations?

A general solution to a system of equations is a set of equations that satisfies all the given equations in the system. It represents all possible solutions to the system and can be expressed using variables or parameters.

2. How is a general solution different from a particular solution?

A particular solution is a specific set of values for the variables in the general solution that satisfies all the given equations in the system. It is a unique solution to the system, while a general solution represents all possible solutions.

3. What is an initial value problem?

An initial value problem is a type of differential equation that involves finding a function that satisfies a given equation and also satisfies a set of initial conditions. These initial conditions provide the starting point for finding a solution to the equation.

4. How do you find a general solution to an initial value problem?

To find a general solution to an initial value problem, you first solve the equation to find a general solution that satisfies the given equation. Then, you use the initial conditions to determine the specific values for the variables in the general solution, resulting in a particular solution that satisfies both the equation and the initial conditions.

5. Why is it important to find a general solution to a system of equations and initial value problem?

Finding a general solution to a system of equations and initial value problem allows us to find all possible solutions to the system and understand the behavior of the system over time. It also helps us to make predictions and analyze the system in different scenarios, making it an essential tool in many fields of science and engineering.

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