University of Florida/Egm4313/s12.team11.imponenti/R3.8

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Problem 3.8[edit]

solved by Luca Imponenti

Kreyszig 2011 pg.84 problem 5[edit]

Problem Statement[edit]

Find a (real) general solution. State which rule you are using. Show each step of your work.

Homogeneous Solution[edit]

To find the homogeneous solution, , we must find the roots of the equation

We know the homogeneous solution for the case of a double root to be

Particular Solution[edit]

We have the following excitation

From table 2.1, K 2011, pg. 82, we have

Since this does not correspond to our homogeneous solution we can use the Basic Rule (a), K 2011, pg. 81 to solve for the particular solution

where

and

Plugging these equations back into the differential equation

from the above equation it is obvious that and

therefore the particular solution to the differential equation is

General Solution[edit]

The general solution will be the summation of the homogeneous and particular solutions

   

The coefficients and can be readily solved for given either initial conditions or boundary value conditions.

Kreyszig 2011 pg.84 problem 6[edit]

Problem Statement[edit]

Find a (real) general solution. State which rule you are using. Show each step of your work.

Homogeneous Solution[edit]

To find the homogeneous solution, , we must find the roots of the equation

with

We know the homogeneous solution for the case of a double root to be

Particular Solution[edit]

We have the following excitation

From table 2.1, K 2011, pg. 82, we have

Since this corresponds to our homogeneous solution we must use the Modification Rule (b), K 2011, pg. 81 to solve for the particular solution

so

differentiating

and

grouping cosine and sine terms we get

and

next we substitute the above equations into the ODE

after cancelling terms; we can equate cosine and sine coefficients to get two equations

so and

and the particular solution to the ODE is

General Solution[edit]

The general solution will be the summation of the homogeneous and particular solutions

   

Egm4313.s12.team11.imponenti 04:28, 21 February 2012 (UTC)