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### Messages - Tim Mengzhe Geng

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16
##### Final Exam / Re: FE-P1
« on: April 11, 2018, 11:15:27 PM »
Actually Professor I remember on the test the last term is -5y^4 instead of -5y^3. I'm not sure whether I'm correct.

It is $-5y^3$.
Yes you are right and the solution is corresponding to the case $-5y^3$.

17
##### Final Exam / Re: FE-P2
« on: April 11, 2018, 11:14:33 PM »
First we find the solution for the homogeneous system

y^{(3)}-2y^{(2)}+4y^{(1)}-8y=0

The corresponding characteristic equation is

r^3-2r^2+4r-8=0

Three roots are

r_1=2

r_2=2i

r_3=-2i

Then the solution for the homogeneous system is

y_c(t)=c_1e^{2t}+c_2\cos(2t)+c_3\sin(2t)

Then we follow to find a particular solution $Y(t)$
We should have

Y(t)=Y_1(t)+Y_2(t)

where

Y_1(t)=Ate^{2t}

and

Y_2(t)=M\sin(t)+N\cos(t)

By plugging in to the equation, we can find that $A=2$, $M=2$ and $N=-4$
In this way we get the required general solution

y(t)=c_1e^{2t}+c_2\cos(2t)+c_3\sin(2t)+2te^{2t}+2\sin(t)-4\cos(t)

18
##### Final Exam / Re: FE-P1
« on: April 11, 2018, 10:51:53 PM »
Actually Professor I remember on the test the last term is -5y^4 instead of -5y^3. I'm not sure whether I'm correct.

19
##### Final Exam / Re: FE-P1
« on: April 11, 2018, 10:50:49 PM »
First we find the integrating factor.
Note that

M_y=2x\cdot\cos(y) - 2xy\cdot\sin(y) - 2y\cdot\cos(x)

N_x=4x\cdot\cos(y) - 2xy\cdot\sin(y) - 3y\cdot\cos(x)

N_x - M_y=2x\cdot\cos(y) -  y\cdot\cos(x)

(N_x - M_y)/M=1/y

Therefore, the integrating factor is only dependent on y.

\ln u= \ln y

u(y)= y

Multiply u(y) on both sides of the equation. Then we have

\phi_x=2xy^2\cos(y) - y^3\cos(x)

\phi=x^2y^2\cos(y)  -y^3\sin(x) +h(y)

\phi_y=2x^2y\cos(y) -x^2y^2\sin(y) - 3y^2\sin(x) +h^\prime(y)

By comparison, we get

h^\prime(y)=-5y^4

h^\prime(y)=-y^5

Then we have

\phi=x^2y^2\cos(y)-y^3\sin(x) -y^5

The general solution is

\phi=x^2y^2\cos(y)-y^3\sin(x) -y^5=c

20
##### MAT244--Misc / Mark for quiz 5
« on: March 15, 2018, 12:12:32 PM »
I wonder whether mark for quiz 5 has been posted on the Blackboard?
I ask this because I take this quiz in a different tutorial session and haven't seen my mark for it on Blackboard yet.

21
##### MAT244--Misc / Term Test 1 Grading
« on: March 06, 2018, 02:26:00 AM »
I have some doubt on the grading of Problem 2.
I guess the form of the solution, prove it works and get the answer right, but finally get no mark for part (a) and half mark for part (b).
May I know which TA I can turn to to discuss my solution?