Toronto Math Forum
MAT2442018S => MAT244––Home Assignments => Web Bonus Problems => Topic started by: Victor Ivrii on January 20, 2018, 09:49:36 AM

a. Find the general solution of
$$
y'+2\cot(t)y=\cos(t).
$$
b. Find solution, which is defined on $(\infty,\infty)$;
c. Find solution, which is defined on $(\infty,\infty)$ and periodic.

Here is my attempt, but I'm confusing about question b and c, is my thought right?
(http://i65.tinypic.com/2qiaccw.jpg)

Here, the last one has the wrong image code
(http://i67.tinypic.com/k3nip3.png)

Looks OK. Could you type it?

My (typed) solution:
(https://i.imgur.com/7MjaNpE.png)

Hi, here is my solution
(https://imgur.com/pdhr4hA.jpg)

(http://i68.tinypic.com/2hxufx2.jpg)
(http://i66.tinypic.com/33ligzk.jpg)

Hello, here is my solution.
(https://i.imgur.com/bLGelce.jpg)

Here is my final answer for week 3 bonus problem(https://imgur.com/00ZZCd1.jpg)

(https://i.imgur.com/g9JMQHW.jpg)

(https://imageshack.com/a/img924/3554/wVaoth.png)

(http://forum.math.toronto.edu/Users/jasonbai/Desktop/53836618264__B3129D997236457691D9553471D77C98.png)
my solution to week3's problem

(https://imgur.com/o77o44u.jpg)

Here is my solution.
(https://i.imgur.com/URLxr3i.jpg)

(https://scontent.fybz21.fna.fbcdn.net/v/t35.012/27265677_10213528247031936_1305947941_o.jpg?oh=a95e4ef8e7887a9b22b5e2746fad679a&oe=5A68D221)

a. y=sint/3+c/(sint)^2
b. y=sint/3
c. y=sint/3

The attached is my solution. Thank you
(https://s17.postimg.org/vv9btu3in/We_Chat_Image_20180122220512.jpg)
(https://s17.postimg.org/3wjrgk6nj/We_Chat_Image_20180122220550.jpg)

The final answer is:(detail steps are in attachment)
a. y=1/3*sint+c*[sin(t)]^(2)
b. When c=0, y=1/3*sin(t)
c. When y=1/3*sin(t), definable on negative infinity to positive infinity, and periodic of 2∏, which is 2 times of Pi.
(https://imgur.com/ucHek9H.jpg)

(https://imgur.com/ArUOkzh.jpg)

(https://imgur.com/a/Hy9lR.jpg)

(https://i.imgur.com/DM9G6cf.jpg?1)

a)
y′+2cot(t)y=cos(t).
p(t)=2cot(t)
I=e^∫2cot(t)dt
=e^2∫cos(t)/sin(t)dt
=e^2lnsin(t)
=e^lnsin(t)^2
=sin(t)^2
sin(t)^2y′+2cot(t)sin(t)^2y=cos(t)sin(t)^2
(sin(t)^2y)'=cos(t)sin(t)^2
sin(t)^2y=∫cos(t)sin(t)^2dt
=1/3sin(t)^3+c
y=1/3sin(t)+csin(t)^2
b)let c=0 y=1/3sin(t)
c)let c=0 y=1/3sin(t)

(http://i66.tinypic.com/2liiyvc.jpg)

(https://imgur.com/VjEW0Xd.jpg)

(https://imgur.com/a/BPEGJ.jpg)

(http://i1347.photobucket.com/albums/p709/shigevertas/WechatIMG2_zpsi8ssxkgk.jpeg)

(http://i1259.photobucket.com/albums/ii541/thhsxhl/IMG_0287_zpssrjwdgfd.jpg)

I already uploaded my solution in reply#30. Sorry for posting the message twice...

(http://i1378.photobucket.com/albums/ah115/tobejy/WechatIMG225_zpshwfpnztr.jpeg)

(https://imgur.com/j39Vs4Q.jpg)

(https://imgur.com/a/EvrWX.jpg)

(https://imgur.com/XePaJ6m.jpg)

Hi, here's my answers to bonus questions. Thank you~
(https://outlook.office.com/owa/service.svc/s/GetAttachmentThumbnail?id=AAMkADQ2NGRjZDJmLWNiYmMtNDg4NS04OWZjLWZiMGRmYzQ3NTQwMQBGAAAAAABKkUHOttDIS7GLrReQieiaBwAbaaUxkwvSSaJMW7e%2BAASOAAAAAAEMAAAbaaUxkwvSSaJMW7e%2BAASOAACsGrlyAAABEgAQAFWUnEwRZJxFhZ3h7FPQMqM%3D&thumbnailType=2&XOWACANARY=Lg5_zdQI1kmUlXZgqtf8A7NL0fYtUYy3wcFV4wGlNw0ZaGbxm5sz8uitDiqjs7RDJUB0fsu8.jpg)

(https://mailattachment.googleusercontent.com/attachment/u/0/?ui=2&ik=b4e13e569d&view=att&th=161216a84bb00cf0&attid=0.1&disp=safe&zw&saddbat=ANGjdJ_EL2qZUslCC_MXkKKNkcLMtsB3NRPLBDQ4n6NQgHYABEe7RfbYaBc0irL12hiiSjsTimGrjwxrmhojlUNv9r0zvhoWTf0NHQH9kNelzuhHxsr4IkdNKZ45lCu5OyiZmxGXJ9mlKCQLDCIoNuLsdcG6HdUCCQDqnh07mLHjx0WZhc4N1zVoGg5FhjcTeykpDwIFv2iLvyuuZNdQ1cYIuU66hQ_ydF6idDUHZFARBUnmK77vRo4swbHyZUuTX_T3LKKKY_LlkWj6WTjMqQMVwFEWvoDO527s4OaXDc7SOeE5nzeNnIC65FTWuQ9WLV3s5IgG3h447uAM6jMkS_qsMUVwRuECUOKLE5YDGGZKGOhbIu3_M7OZhcP_0D4lBsjqgZF__hojLYIZ8m0NM4H2on6DfKj59YO7mQq3O1DAoDWecxB0iZ82is8b6TpqTpX4QbCY4OpiZRNtEJygk4UiQoaVCCYz756vwn0anuU_mM3xe6ESXqC5ieJfbmokcZRFnq7OyedLbNgPlULjM4Ljth_kqTbLLbVHMbSHXy1ETdB48i9ANvGkpMBFPfYUz3AMuopvngkOqfVvLgvaCJykDGQ，https://mail.google.com/mail/u/0/)

(https://imgur.com/WKjMnvo.jpg)

(https://imgur.com/QNKVZ8Z.jpg)
(https://imgur.com/q0S6NXY.jpg)

(https://s9.postimg.org/4hmh2obsf/1520676482.jpg)

(http://i1036.photobucket.com)

(http://[URL=http://s1036.photobucket.com/user/YulingChen/media/20180123_zpsqewy3fyx.png.html][IMG]http://i1036.photobucket.com/albums/a443/YulingChen/20180123_zpsqewy3fyx.png)[/URL][/img]

(http://i1036.photobucket.com/albums/a443/YulingChen/20180123_zpsqewy3fyx.png)

(https://imgur.com/k3i3yZM.jpg)

Here is the solution for the question:
(https://photos.smugmug.com/Week3/iQM3NnJq/0/99585bea/XL/week3bounsXL.png)

(http://i1356.photobucket.com/albums/q722/hyylor/IMG_0838_zpsx1s4kgbr.png) (http://s1356.photobucket.com/user/hyylor/media/IMG_0838_zpsx1s4kgbr.png.html)

here is my solution
(https://i.imgur.com/UNCIH34.jpg)

(https://imgur.com/2aaczTj.jpg)

When I wrote "type" I meant type on forum. Like I do:
$$y'+2\cot(t)y=\cos(t)$$,
then
$$I=\exp\Bigl(2\cot(t)\,dt\Bigr)=\exp \bigl(2\ln \sin(t)\bigr)=\sin^2(t)$$
and so on, not to put image (even typed) on forum.
Don't cut lines too short... Also $\exp $, $\cos$ etc should be upright (in $\LaTeX$ it is \exp, \cos).
No external images! Post attachments

Usually web bonus is given to the first poster (or may be for improvements if the first poster did not get everything right). This time I was more generous. But not infinitely. Bonus got first 5 or so