What Are the Phase Lines and Equilibrium Points for These Differential Systems?

[MrBioMedic]

Hello everyone - I have a couple questions about some homework problems that I am hoping I can get some help with - Here they are and thank you for the help in advance.


for the system dY / dt = | a 1 | (2 by 2 matrix)
----------------------- | 0 -1 |


and the system dY / dt = | a 1 |
------------------------- | -1 2 |



I need to figure out the following:
a. possible phase lines for each as a varies.
b. classify the equilbrium points
c. the general solution for a=2


Thank you again for any help.

 

[ReyChiquito]

For the problem

$$\frac{d \vec{y}}{dt}=\bold{A}\vec{y}$$

where $\bold{A}$ is a matrix of constant coefficients, you know that the general solution for the system is
$$\vec{y}(t)=\vec{y_{0}}e^{\bold{A}t}$$

so, here is the canonical way to go...

diagonalize $\bold{A}$. You will get your solution to be in the form*
$$\vec{z}(t)=\left(\begin{array}{cc}z_{01}e^{\lambda_{1}t}\\z_{02}e^{\lambda_{2}t}\end{array}\right)$$

Where $\vec{y}=\bold{P}\vec{z}$ and $P$ is the rotation matrix formed by the eigenvectors of [itex]\bold{A}[/tex], and $\lambda_{1},\lambda_{2}$ are the eigenvalues.<br /> <br /> So now you can express $y=y(x,\lambda_{1},\lambda_{2})$ and see the form of the integral paths in the phase plane.<br /> <br /> Now, the equilibrium points will be classified by these integral curves. The direction of the paths in this points is given by the eigenvectors.<br /> <br /> * This will be the form of the solution <b>only</b> when $\bold{A}$ is similar to a diagonal matrix. If not, the matrix will be similar to a Jordan block, and your solution will change (for further reference check almost any ode book), and i think you will have to do that case also.<br /> <br /> PS. sorry for bad english.<br /> <br /> TO MODS. The Spellcheck should skip text between $$tags... that would be nice.. just an idea <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":smile:" title="Smile :smile:" data-smilie="1"data-shortname=":smile:" />$$[/itex]

 

[M98Ranger]

Hi there! I'd be happy to help with your differential homework questions. Let's take a look at each of the problems and break down how to approach them.

For the first system, we have the equation dY / dt = | a 1 | (2 by 2 matrix)
----------------------- | 0 -1 |
To determine the possible phase lines for each value of a, we can use the eigenvalues of the matrix. The eigenvalues of a 2 by 2 matrix are given by the formula λ = (a - d) ± √((a - d)² + 4bc) / 2, where a, b, c, and d are the values in the matrix. In this case, we have a = a, b = 1, c = 0, and d = -1.

For the first system, we have a phase line that looks like this:

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- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

For the second system, we have the equation dY / dt = | a 1 | (2 by 2 matrix)
----------------------- | -1 2 |
Using the same formula, we can find the eigenvalues to be λ = (a - d) ± √((a - d)² + 4bc) / 2. In this case, we have a = a, b = 1, c = -1, and d = 2.

For the second system, we have a phase line that looks like this:

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -