Characteristic Equations and Solutions

This video tutorial covers solving constant coefficient second order linear homogeneous differential equations. It begins with an introduction to the equations and proceeds to solve the characteristic equation using examples. The tutorial explains how to find linearly independent solutions and determine constants using initial conditions. It discusses different cases of roots, including real, repeated, and complex roots, and provides examples for each. The video concludes with a summary and final notes.

What is the form of a constant coefficient second order linear homogeneous differential equation?

• a times y double prime plus b times y prime plus c times y equals zero

• a times y prime plus b times y equals zero

• a times y prime plus b times y double prime plus c equals zero

• a times y double prime plus b times y equals zero

In the example problem, what are the values of a, b, and c?

• a = 1, b = -6, c = 8

• a = 1, b = 6, c = -8

• a = 2, b = -6, c = 8

• a = 1, b = -8, c = 6

What is the characteristic equation derived from the example problem?

• r squared plus eight R minus six equals zero

• r squared minus eight R plus six equals zero

• r squared plus six R minus eight equals zero

• r squared minus six R plus eight equals zero

What is the general solution form when the characteristic equation has distinct real roots?

• y of x equals c sub one times e to the power of R sub One X plus C sub two times e to the power of R sub two x

• y of x equals c sub one times e to the power of R sub One X

• y of x equals c sub one times e to the power of R sub two x

• y of x equals c sub one times e to the power of R sub One X plus C sub two

What additional factor is included in the general solution when the characteristic equation has repeated real roots?

• An extra factor of e

• An extra factor of X

• An extra factor of R

• An extra factor of C

What is the characteristic equation for the example with repeated real roots?

• r squared minus eight R plus 16 equals zero

• r squared plus eight R minus 16 equals zero

• r squared plus 16 R minus eight equals zero

• r squared minus 16 R plus eight equals zero

What is the form of the general solution when the characteristic equation has complex roots?

• y of x equals c sub one e to the power of alpha x cosine beta X plus C sub two e to the power of alpha x times sine of beta X

• y of x equals c sub one e to the power of beta x sine alpha X

• y of x equals c sub one e to the power of alpha x

• y of x equals c sub one cosine beta X plus C sub two sine of beta X

In the example with complex roots, what are the values of alpha and beta?

• Alpha is 4 and beta is 3

• Alpha is 3 and beta is 4

• Alpha is 2 and beta is 3

• Alpha is 3 and beta is 2

What is the particular solution for the example with complex roots?

• y equals five times e to the power of three x sine 2X

• y equals five times e to the power of two x sine 3X

• y equals five times e to the power of three x cosine 2X

• y equals five times e to the power of two x cosine 3X

What method can be used to solve the characteristic equation with complex roots?

• Using the factorization method

• Using the quadratic formula only

• Completing the square or using the quadratic formula

• Completing the square only