Other
APM2611 ASSIGNMENT SOLUTIONS 2021 SEM1
- Course
- Institution
This document contains APM2611 Assignment 1 Solutions(Latest tutorial letter). All working are shown clearly and explanations are provided when necessary
[Show more]
Seller
Follow
jctutor0814378595
Reviews received
Content preview
APM2611ASSIGNMENT 1 2021
QUESTION 1
QUESTION 1.1
4𝑧 ′′ − 4𝑧 ′ − 3𝑧 = cos(2𝑥)
𝐶𝑜𝑚𝑝𝑙𝑖𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑧𝑐 :
4𝑚2 − 4𝑚 − 3 = 0 ∴ 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
(2𝑚 + 1)(2𝑚 − 3) = 0
1 3
𝑚1 = − 𝑎𝑛𝑑 𝑚2 =
2 2
𝑧𝑐 = 𝐶1 𝑒 𝑚1 𝑥 + 𝐶2 𝑒 𝑚2 𝑥
𝑥⁄ 3𝑥⁄
𝑧𝑐 = 𝐶1 𝑒 − 2 + 𝐶2 𝑒 2
𝑃𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑧𝑝 :
𝐿𝑒𝑡: 𝑧𝑝 = 𝐴 + 𝐵 cos(2𝑥)
𝑧𝑝 = 𝐴 sin(2𝑥) + 𝐵 cos(2𝑥)
′
(𝑧𝑝 ) = 2𝐴 cos(2𝑥) − 2𝐵 sin(2𝑥)
′′
(𝑧𝑝 ) = −4𝐴 sin(2𝑥) − 4𝐵 cos(2𝑥)
4𝑧 ′′ − 4𝑧 ′ − 3𝑧 = cos(2𝑥)
′′ ′
4(𝑧𝑝 ) − 4(𝑧𝑝 ) − 3𝑧𝑝 = cos(2𝑥)
4[−4𝐴 sin(2𝑥) − 4𝐵 cos(2𝑥)] − 4[2𝐴 cos(2𝑥) − 2𝐵 sin(2𝑥)] − 3[𝐴 sin(2𝑥) + 𝐵 cos(2𝑥)]
= cos(2𝑥)
−16𝐴 sin(2𝑥) − 16𝐵 cos(2𝑥) − 8𝐴 cos(2𝑥) + 8𝐵 sin(2𝑥) − 3𝐴 sin(2𝑥) − 3𝐵 cos(2𝑥) = cos(2𝑥)
(−16𝐴 + 8𝐵 − 3𝐴) sin(2𝑥) + (−16𝐵 − 8𝐴 − 3𝐵) cos(2𝑥) = cos(2𝑥)
(−19𝐴 + 8𝐵) sin(2𝑥) + (−19𝐵 − 8𝐴) cos(2𝑥) = cos(2𝑥)
,−19𝐴 + 8𝐵 = 0 1
−8𝐴 − 19𝐵 = 1 2
19𝐴
𝐹𝑟𝑜𝑚 1 𝐵=
8
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝐵 𝑖𝑛𝑡𝑜 2
−8𝐴 − 19𝐵 = 1 2
19𝐴
−8𝐴 − 19 ( )=1
8
361𝐴
−8𝐴 − =1
8
−64𝐴 − 361𝐴 = 8
−425𝐴 = 8
8
𝐴=−
425
19𝐴
𝐵=
8
19 8 19
𝐵= (− )=−
8 425 425
𝑧𝑝 = 𝐴 sin(2𝑥) + 𝐵 cos(2𝑥)
8 19
𝑧𝑝 = − sin(2𝑥) − cos(2𝑥)
425 425
𝑧 = 𝑧𝑐 + 𝑧𝑝
𝒙⁄ 𝟑𝒙⁄ 𝟖 𝟏𝟗
𝒛 = 𝑪𝟏 𝒆− 𝟐 + 𝑪𝟐 𝒆 𝟐 − 𝐬𝐢𝐧(𝟐𝒙) − 𝐜𝐨𝐬(𝟐𝒙)
𝟒𝟐𝟓 𝟒𝟐𝟓
QUESTION 1.2
𝑥 2 𝑦 ′′ − 3𝑥𝑦 ′ + 3𝑦 = 12𝑥 4
, 3 3
𝑦 ′′ − 𝑦 ′ + 2 𝑦 = 12𝑥 4
𝑥 𝑥
𝐺𝑖𝑣𝑒𝑛 ∶ 𝑦1 = 𝑥 𝑎𝑛𝑑 𝑦2 = 𝑥 3
𝑦𝑝 = 𝑢1 𝑦1 + 𝑢2 𝑦2
𝑊 = |𝑥 𝑥 3 | = 3𝑥 3 − 𝑥 3 = 2𝑥 3
1 3𝑥 2
3
| 𝑥 2 𝑥 2 | 0 − 12𝑥 5
(𝑢1 )′ = 12𝑥 33𝑥 = = −6𝑥 2
2𝑥 2𝑥 3
−6𝑥 3
𝑢1 = ∫ −6𝑥 2 𝑑𝑥 = + 𝐶1 = −2𝑥 3 + 𝐶1
3
𝑥 0
| | 3 3
(𝑢2 )′ = 1 12𝑥 2 = 12𝑥 − 0 = 12𝑥 = 6
2𝑥 3 2𝑥 3 2𝑥 3
𝑢2 = ∫ 6 𝑑𝑥 = 6𝑥 + 𝐶2
𝑦𝑝 = 𝑢1 𝑦1 + 𝑢2 𝑦2
𝑦𝑝 = (−2𝑥 3 + 𝐶1 )𝑥 + (6𝑥 + 𝐶2 )𝑥 3
𝑦𝑝 = −2𝑥 4 + 𝐶1 𝑥 + 6𝑥 4 + 𝐶2 𝑥
𝒚𝒑 = 𝑪𝟏 𝒙 + 𝑪𝟐 𝒙 + 𝟒𝒙𝟒
QUESTION 2
𝑅𝑖𝑛 = 𝑖𝑛𝑝𝑢𝑡 𝑟𝑎𝑡𝑒
𝑅𝑜𝑢𝑡 = 𝑜𝑢𝑡𝑝𝑢𝑡 𝑟𝑎𝑡𝑒
𝑁 = 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑠𝑎𝑙𝑡 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡.
𝑑𝑁
= 𝑅𝑖𝑛 − 𝑅𝑜𝑢𝑡
𝑑𝑡
1𝑔 4𝐿
𝑅𝑖𝑛 = × = 4 𝑔⁄𝑚𝑖𝑛
𝐿 𝑚𝑖𝑛
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jctutor0814378595. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $5.84. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
85443 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now