Assignment 17

#$&*

course Mth 279

7/15

Query 15 Differential Equations*********************************************

Question: Suppose y1 and y2 are solutions to y '' + 2 t y ' + t^2 y = 0. If y1(3) = 0, y1 ' (3) = 0, y2(3) = 1 and y2 ' (3) = 2, can you say whether {y1, y2} is a fundamental set? If so, is it or isn't it?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Use Wronskian

W(t) = |0 0 ; 1 2| = (0*2) - (0*1) = 0

No fundamental set of {y_1,y_2}

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

1

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

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

Self-critique rating:

*********************************************

Question: Are y1 = 2 e^(-2 t) cos(t) and y2 = e^(-2 t) sin(t) solutions to the equation

y '' + 4 y ' + 5 y = 0?

What are the initial conditions at t = 0?

Is {y1, y2} a fundamental set?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

A) y_1 = 2e^(-2t)*cos(t)

y’_1 = -2e^(-2t)sin(t) - 4e^(-2t)cos(t)

y’’_1 = -2e^(-2t)cos(t) + 4e^(-2t)sin(t) + 8e^(-2t)cos(t) - 4e^(-2t)(-sin(t))

y’’_1 = 6e^(-2t)cos(t) + 8e^(-2t)sin(t)

plugging values in

(6e^(-2t)cos(t) + 8e^(-2t)sin(t)) + 4(-2e^(-2t)sin(t) - 4e^(-2t)cos(t)) + 5(2e^(-2t)cos(t)) =? 0

yes does = 0

y_2 = e^(-2t)sin(t)

y’_2 = -2e^(-2t)sin(t) + e^(-2t)cos(t)

y’’_2 = 3e^(-2t)sin(t) - 4e^(-2t)cos(t)

plugging in it equals 0

So both y_1 and y_2 are solutions of the equation

B) initial conditions at t=0

y_1(0) = 2e^(-2*0)cos(0) = 2

y_1(0)=2

y’_1(0) = -2e^(-2*0)sin(0) - 4e^(0)cos(0) = -4

y’1(0) = -4

y_2 (0) = e^(0)sin(0) = 0

y_2(0) = 0

y’_2(0) = -2e^0sin(0) + e^0cos(0) = 1

y’_2(0) = 1

C) do Wronskian to see if fundamental set

W(t) = |2 0; -4 1| = 2(1) - 0(-4) = 2

Yes {y_1 , y_2} are fundamental set

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

1

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

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

Self-critique rating:

*********************************************

Question: y1_bar = 2 y1 - 2 y2 and y2_bar = y1 - y2. Is {y1_bar, y2_bar} a fundamental set?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Use Wronskian but first need to find y’_1bar and y’_2bar

y1_bar = 2y1 - 2y2

y’1_bar = 2y’_1 - 2y’_2

y2_bar = y_1 - y_2

y’2_bar = y’_1 - y’_2

W(t) = |2y1 - 2y2, y_1 - y_2; 2y’_1 - 2y’_2, y’_1 - y’_2|

= (2y_1 - 2y_2)(y’_1 - y’_2) - (y_1 - y_2)(2y’_1 - 2y’_2) = 0

Not fundamental set

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

1

.............................................

Given Solution: Note that y_1_bar = 2 * y_2_bar.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

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

Self-critique rating:

*********************************************

Question: Is {e^t, 2 e^(-t), sinh (t) } a fundamental set on the interval (-infinity, infinity)?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

sinh(t) = (e^t - e^-t)/2

W(t) = |y_1, y_2, y_3; y’1, y’2, y’3; y’’1, y’’2, y’’3|

Not sure what W(t) equals in a 3x3 matrix in general form

@&

You just keep adding higher-order derivatives:

The Wronskian of the 3-element set

{y_1, y_2, y_3}

is

det ( [ y_1, y_2, y_3; y_1 ' , y_2 ', y_3 '; y_1 '', y_2 '', y_3 ''] ).

*@

&#Good responses. See my notes and let me know if you have questions. &#