Lecture14(7.5,7.6).pdf

# Find a fundamental set for the system a im working b

• Notes
• 39

This preview shows page 31 - 39 out of 39 pages.

Find a fundamental set for the system. A I’m working B I’m done C I’m stuck

Subscribe to view the full document.

Consider the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Find a fundamental set for the system. A I’m working B I’m done C I’m stuck HINT: The eigenvalues/vectors: r 1 = - 1 1 - 1 r 2 = - 2 1 - 2
Let’s consider the associated homogeneous equations: x 00 + 3 x 0 + 2 x = 0 which corresponds to the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Any vector-valued function of the following form is a solution: ~ x ( t ) = c 1 e - t 1 - 1 + c 2 e - 2 t 1 - 2

Subscribe to view the full document.

Let’s consider the associated homogeneous equations: x 00 + 3 x 0 + 2 x = 0 which corresponds to the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Any vector-valued function of the following form is a solution: ~ x ( t ) = c 1 e - t 1 - 1 + c 2 e - 2 t 1 - 2 Notice that these solutions correspond to the solutions we would have gotten solving the 2nd order equation.
Let’s consider the associated homogeneous equations: x 00 + 3 x 0 + 2 x = 0 which corresponds to the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Any vector-valued function of the following form is a solution: ~ x ( t ) = c 1 e - t 1 - 1 + c 2 e - 2 t 1 - 2 Notice that these solutions correspond to the solutions we would have gotten solving the 2nd order equation. Solutions tend to 0 as t → ∞ .

Subscribe to view the full document.

Let’s consider the associated homogeneous equations: x 00 + 3 x 0 + 2 x = 0 which corresponds to the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Any vector-valued function of the following form is a solution: ~ x ( t ) = c 1 e - t 1 - 1 + c 2 e - 2 t 1 - 2 Notice that these solutions correspond to the solutions we would have gotten solving the 2nd order equation. Solutions tend to 0 as t → ∞ . The equilibrium x = 0 is asymptotically stable.
Let’s consider the associated homogeneous equations: x 00 + 3 x 0 + 2 x = 0 which corresponds to the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Any vector-valued function of the following form is a solution: ~ x ( t ) = c 1 e - t 1 - 1 + c 2 e - 2 t 1 - 2 Notice that these solutions correspond to the solutions we would have gotten solving the 2nd order equation. Solutions tend to 0 as t → ∞ . The equilibrium x = 0 is asymptotically stable. It is called an asymptotically stable node.

Subscribe to view the full document.

Consider the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x .

{[ snackBarMessage ]}

###### "Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern