Query 15

#$&*

course MTH 279

3/11 8 pm

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: Yes, you can use the fact that the sum of y_1 and y_2 isn't equal to 0 at a particular t value to verify that they are a fundamental set. Yes, they are a fundamental set. I think this statement hold even though the computed Wronskian is 0.

@& If the Wronskian is zero at a point in an interval of the domain, then it is zero at all points of the domain.

In this case the domain over which the solution function exists is the entire real line, so the Wronskian is zero at all points, and it follows that {y_1, y_2} cannot be a solution set. *@

confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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

Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK

********************************************* 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: Yes, both y_1 and y_2 are solutions. And yes, they create a fundamental set since their sum evaluated at t = 0 isn't 0. y_1(0) = 2, y_1'(0) = -4, y_2(0) = 0 , y_2'(0) = 1

@& The Wronskian at zero is det ( [2, 0; -4, 1] ) = 2, which is nonzero. The set is therefore a fundamental set. *@

confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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

Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK

********************************************* Question: y1_bar = 2 y1 - 2 y2 and y2_bar = y1 - y2. Is {y1_bar, y2_bar} a fundamental set? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: No, the equations are constant multiples of eachother and therefore do not form a fundamental set.

@& Correct.

You should also verify that the Wronskian is zero. This is a simple matter of writing it out, and will reinforce the connection between linear independence and the Wronskian. *@

confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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

Given Solution: Note that y_1_bar = 2 * y_2_bar. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK

********************************************* Question: Is {e^t, 2 e^(-t), sinh (t) } a fundamental set on the interval (-infinity, infinity)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Yes. The sum of the equations is (3/2)(e^t + e^-t), which gives you a nonzero answer for any value of t that you may plug in. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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

Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): 3 ------------------------------------------------ Self-critique rating: OK"

@& It isn't difficult to find a linear combination of these three functions that is identically zero. For example, the first minus half the second, minus double the third is, I believe, identically zero.

You should also evaluate the Wronskian and see that it is zero. Once more this reinforces the essential connections. *@

Self-critique (if necessary): ------------------------------------------------ Self-critique rating:

Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!

@& You need to rely more on the Wronskian, and see how it is connected to the linear independence or dependence of the various sets.

Check my notes. If you do as I suggest everything should become clear in fairly short order. *@

Query 15

#$&*

course MTH 279

3/11 8 pm

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:

Yes, you can use the fact that the sum of y_1 and y_2 isn't equal to 0 at a particular t value to verify that they are a fundamental set. Yes, they are a fundamental set.

I think this statement hold even though the computed Wronskian is 0.

@&

If the Wronskian is zero at a point in an interval of the domain, then it is zero at all points of the domain.

In this case the domain over which the solution function exists is the entire real line, so the Wronskian is zero at all points, and it follows that {y_1, y_2} cannot be a solution set.

*@

confidence rating #$&*:

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

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

Given Solution:

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

Self-critique (if necessary): OK

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

Self-critique rating: OK

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

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:

Yes, both y_1 and y_2 are solutions. And yes, they create a fundamental set since their sum evaluated at t = 0 isn't 0.

y_1(0) = 2, y_1'(0) = -4, y_2(0) = 0 , y_2'(0) = 1

@&

The Wronskian at zero is det ( [2, 0; -4, 1] ) = 2, which is nonzero. The set is therefore a fundamental set.

*@

confidence rating #$&*:

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

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

Given Solution:

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

Self-critique (if necessary): OK

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

Self-critique rating: OK

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

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

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

No, the equations are constant multiples of eachother and therefore do not form a fundamental set.

@&

Correct.

You should also verify that the Wronskian is zero. This is a simple matter of writing it out, and will reinforce the connection between linear independence and the Wronskian.

*@

confidence rating #$&*:

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

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

Given Solution: Note that y_1_bar = 2 * y_2_bar.

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

Self-critique (if necessary): OK

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

Self-critique rating: OK

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

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

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Yes. The sum of the equations is (3/2)(e^t + e^-t), which gives you a nonzero answer for any value of t that you may plug in.

confidence rating #$&*:

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

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

Given Solution:

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

Self-critique (if necessary): 3

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

Self-critique rating: OK"

@&

It isn't difficult to find a linear combination of these three functions that is identically zero. For example, the first minus half the second, minus double the third is, I believe, identically zero.

You should also evaluate the Wronskian and see that it is zero. Once more this reinforces the essential connections.

*@

Self-critique (if necessary):

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

Self-critique rating:

Self-critique (if necessary):

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

Self-critique rating:

#*&!

@&

You need to rely more on the Wronskian, and see how it is connected to the linear independence or dependence of the various sets.

Check my notes. If you do as I suggest everything should become clear in fairly short order.

*@