Query 02

#$&*

course Mth 279

10/8 noon

Solve each equation:*********************************************

Question: 1. y ' + y = 3

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p(t)=1 integral 1=t

y=Ce^(-t)+3

confidence rating #$&*:ut of 3

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

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

Given Solution:

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

Self-critique (if necessary):

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

Self-critique rating:

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

Question:

2. y ' + t y = 3 t

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p(t)=t integral t=(t^2)/2

u(t)=e^((t^2)/2)

e^((t^2)/2)*y’+t* e^((t^2)/2)*y=3* e^((t^2)/2)

@& The right-hand side is 3 t e^(t^2 / 2).

You can't integrate just e^(t^2 / 2).*@

e^((t^2)/2)*y=3* e^((t^2)/2)+C

y=3+C* e^(-(t^2)/2)

@& e^(t^2 / 2) is not an antiderivative of e^(t^2 / 2). The derivative of e^(t^2 / 2) is 2 t e^(t^2 / 2).

You can't integrate the right-hand side unless you include that t.*@

confidence rating #$&*:ut of 3

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

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

Given Solution:

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

Self-critique (if necessary):

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

Self-critique rating:

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

Question:

3. y ' - 4 y = sin(2 t)

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p(t)=-4 integral -4=-4*t

u(t)=e^-4*t

e^-4t*y’-(4e^-4*t)=sin(2t)(e^-4t)

e^(-4t)*y=4e^(-4t)+sin(2t)(e^-4t)

@& You are following the steps correctly, but you have not correctly integrated sin(2 t) e^(-4 t).

To integrate this you need to integrate by parts and develop a reduction formula.*@

@& You should check to convince yourself that sin(2 t) e^(-4 t) is not an antiderivative of sin(2 t) e^(-4 t)..*@

y=Ce^(4t)+sin(2t)

confidence rating #$&*:ut of 3

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

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

Given Solution:

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

Self-critique (if necessary):

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

Self-critique rating:

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

Question:

4. y ' + y = e^t, y (0) = 2

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p(t)=1 integral 1=t

u(t)=e^t

e^t*y’+e^t*y=(e^t)(e^t)

General Solution: y=Ce^-t+(e^t/2)

@& e^t * e^t = e^(2 t), which then needs to be integrated.*@

C=12.5

Particular Solution: y=12.5e^-2+(e^t/2)

confidence rating #$&*:ut of 3

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

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

Given Solution:

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

Self-critique (if necessary):

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

Self-critique rating:

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

Question:

5. y ' + 3 y = 3 + 2 t + e^t, y(1) = e^2

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p(t)=3 integral 3=3*t

u(t)=e^(3*t)

e^(3*t)*y’+e^(3*t)*y=3e^(3t)+2*(te^(3*t))+(e^(t)*e^(3*t)

General Solution: y=Ce^(-3t)+(2*t/3)+(e^t/4)

@& The right-hand side needs to be integrated correctly.*@

C=121.37

Particular Solution: y=121.37*e^(-3*t)+(2*t/3)+(e^t/4)

confidence rating #$&*:ut of 3

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

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

Given Solution:

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

Self-critique (if necessary):

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

Self-critique rating:

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

Question:

6. The general solution to the equation y ' + p(t) y = g(t) is y = C e^(-t^2) + 1, t > 0. What are the functions p(t) and g(t)?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p(t)=2t

g(t)=1/(e^(t^2))

confidence rating #$&*:ut of 3

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

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

Given Solution:

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

Self-critique (if necessary):

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

Self-critique rating:

"

Self-critique (if necessary):

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

Self-critique rating:

@& You are generally following the correct steps in the solution, but you don't appear to have been integrating the right-hand side.

As a result most of your solutions do not work in the given equations. If you substitute your y functions into the equations and simplify, you'll see what I mean. Your solution does happen to work on the first problem, but on that one you made two errors, one cancelling the other.*@