OpenLLM-France/wikimedia huggingface dataset:
# PowerPC Assembly/Printable version

The current, editable version of this book is available in Wikibooks, the open-content textbooks collection, at
https://en.wikibooks.org/wiki/PowerPC_Assembly

# Introduction
## Why Learn Assembly?

Assembly is the most primitive tool in the programmers toolbox. Entire software projects can be written without ever once looking at a single line of assembly code. So the question arises: why learn assembly? Assembly language is the closest form of communication that humans can engage in with a computer. Using assembly, the programmer can precisely track the flow of data and execution in a program. Also, another benefit to learning assembly, is that once a program has been compiled, it is difficult -- if not impossible -- to decompile the code. That means that if you want to examine a program that is already compiled, you will need to examine it in assembly language. Debuggers also will frequently only show the program code in assembly language. If nothing else, it can be beneficial to learn to read assembly language, if not write it.
Assembly language is also the preferred tool, if not the only tool available for implementing some low-level tasks, such as bootloaders, and low-level kernel components. Code written in assembly has less overhead than code written in high-level languages, so assembly code frequently will run much faster than programs written in other languages. Code that is written in a high-level language can be compiled into assembly, and "hand optimized" to squeeze every last bit of speed out of a section of code. As hardware manufacturers such as Intel and AMD add new features and new instructions to their processors, often times the only way to access those features is to use assembly routines. That is, at least until the major compiler vendors add support for those features.
Developing a program in assembly can be a very time consuming process, however. While it might not be a good idea to write new projects in assembly language, it is certainly valuable to know a little bit about assembly language anyway.

## Who is This Book For?

This book is for those who are learning assembly for the first time, or hoping to optimize their PPC specific code, or even those hoping to learn more about a RISC processor.

# Instructions
## Instruction Forms

All PowerPC instructions are four bytes long. Whenever the processor calculates the destination address of a branch, the two low-order bits are ignored, so the actual two low-order bits are always 0 in the destination address (i.e., every instruction is word-aligned).
Bits 0-5 of an instruction are specified by the op code, and also indicate the form of the instruction (how to divide up the remaining bits of the instruction into fields, and the meaning of each field).

## Further reading

* ↑ "POWER family and PowerPC Architecture Overview : Instruction Forms".
* "Assembler Language Reference for for POWER-based architectures". (includes both Power2 and PowerPC instruction sets).
# Powhatan/Grammar
## Grammar Outlines

   Grammar Outline: ---
       I Morphosyntax
          A Direct
          1 Verbs with 1st or 2nd person subjects and third person objects are direct and are marked with a suffix
          B Inverse
          1 Verbs with 3rd person subjects and 1st or 2nd person objects are inverse and are marked with a suffix
          C Negation
          1 The Powhatan negative prefix “MaTa” has several forms which are in complementary distribution
       II Morphology
          A Noun Inflection (Powhatan has extensive verbal and nominal morphology)
          1 Gender
             a Nouns are divided into two different genders; animate and inanimate. This comes into play with pluralization and possession
          2 Number
             a There are two number distinctions; singular and plural
          3 Obviation
             a When 2 or more animate nouns occur within a sentence and are not conjoined, the focus or proximate noun is inflected for major 3rd person. The remaining nouns are inflected for minor 3rd person
          4 Possession
             a Possessed nouns are inflected for person and number
             b Possessive prefixes can be used with almost any noun in Mattaponi
             c Certain nouns (including most body parts and kinship terms, and some words for personal objects like clothing) have inalienable possession. That means you must use a possessive prefix with one of those words. For these words, the possessive pronouns are slightly different. The pronouns are nih-, kih-, o-, and mih- before a root noun that begins with a consonant, and n-, k-, w-, and m- before a root noun that begins with a vowel.
             d When animate words use the third person form ("his or her"), there is not only a prefix (o- or w-) but also a suffix (-an) at the end of the word. This is not true for inanimate words.
          B Verb Inflection (There are 4 Powhatan verb categories: Inanimate Intransitive (II), Animate Intransitive (AI), Transitive Inanimate (TI), and Transitive Animate (TA))
          1 Intransitive Verbs
             a Agree with the gender of their subjects
          2 Transitive Verbs
             a Agree with the number of their inanimate objects. If the subject noun is not present then the verb also has inflection for the number of the subject. Transitive Animate Verbs
                1 Agree with the number of both their subjects and their objects which are animate
       III Syntax
          A Syllable Stress
          1 syllables alternate in stress from light to heavy
          2 if the 1st vowel is weak (uh or e) then even syllables are light, otherwise odd syllables are light
          B Word Order
          1 Mattaponi's word order is primarily SUBJECT VERB OBJECT (SVO), but there are exceptions
          C Topicalization
          1 This is a property of Powhatan syntax which is achieved by adding an independent pronoun
          D Questions
          1 Yes/No
             a The use of non-affirmative endings alone or in addition to an interrogative prefix.
          2 Content / WH-word Questions
             a Questioned elements are moved to the beginning of the sentence
# Powhatan/Lesson 1

Family Titles (roman)
| Family Member | My | Your | His / Hers | Someone’s |
| Father | Nosh | Kosh | O’Sha | Mosh |
| Mother | Nek | Kek | O’Ka | Mek |
| Son | NuhKwiSuhs | KuhKwiSuhs | Ah’KwiSuhs | MuhKwiSuhs |
| Daughter | NuhNuhTaNuhs | KuhNuhTaNuhs | Ah’NuhTaNuhs | MuhNuhTaNuhs |
| Brother (older) | NiSHauns | KiSHauns | Ah’SHauns | MiSHauns |
| Brother (younger) | NuhMat | KuhMat | O’Mat | MuhMat |
| Sister (older) | NuhMis | KuhMis | O’Mis | MuhMis |
| Sister (younger) | NuhSiMuhs | KuhSiMuhs | Ah’SiMuhs | MuhSiMuhs |
| Husband | Ni’O’Hwa | Ki’O’Hwa | Wi’O’Hwa | Mi’O’Hwa |
| Wife | NiWa | KiWa | WiWa | MiWa |
| Uncle (paternal) | NoSHaWas | KoSHaWas | Ah’SHaWas | MoSHaWas |
| Uncle (maternal) | NuhSHis | KuhSHis | Ah’SHis | MuhSHis |
| Aunt (paternal) | NuhMiDiTauk | KuhMiDiTauk | Ah’MiDiTauk | MuhMiDiTauk |
| Aunt (maternal) | NuhKuhKush | KuhKuhKush | Ah’KuhKush | MuhKuhKush |
| Cousin (male) | NuhTanKwus | KuhTanKwus | Ah’TanKwus | MuhTanKwus |
| Cousin (female) | NuhTanKwuhSosKwo | KuhTanKwuhSosKwo | Ah’TanKwuhSosKwo | MuhTanKwuhSosKwo |
| Friend (inside) | WiNGaPo | WiNGaPo | WiNGaPo | WiNGaPo |
| Friend (outside) | NiTaup | KiTaup | O’Taup | MiTaup |
Family Titles (script)
| Family Member | My | Your | His / Hers | Someone’s |
| Father | ᓄᔥ | ᑯᔥ | ᐅᔕ | ᒧᔥ |
| Mother | ᓀᒃ | ᑫᒃ | ᐅᑲ | ᒣᒃ |
| Son | ᓀᐦᑭᐧᓭᐦᔅ | ᑫᐦᑭᐧᓭᐦᔅ | ᐊᐦᑭᐧᓭᐦᔅ | ᒣᐦᑭᐧᓭᐦᔅ |
| Daughter | ᓀᐦᓀᐦᑕᓀᐦᔅ | ᑫᐦᓀᐦᑕᓀᐦᔅ | ᐊᐦᓀᐦᑕᓀᐦᔅ | ᒣᐦᓀᐦᑕᓀᐦᔅ |
| Brother (older) | ᓂᔕᐠᐦᓐᔅ | ᑭᔕᐠᐦᓐᔅ | ᐊᐦᔕᐠᐦᓐᔅ | ᒥᔕᐠᐦᓐᔅ |
| Brother (younger) | ᓀᐦᒪᑦ | ᑫᐦᒪᑦ | ᐅᒪᑦ | ᒣᐦᒪᑦ |
| Sister (older) | ᓀᐦᒥᔅ | ᑫᐦᒥᔅ | ᐅᒥᔅ | ᒣᐦᒥᔅ |
| Sister (younger) | ᓀᐦᓯᒣᐦᔅ | ᑫᐦᓯᒣᐦᔅ | ᐊᐦᓯᒣᐦᔅ | ᒣᐦᓯᒣᐦᔅ |
| Husband | ᓂᐅᕹᐧ | ᑭᐅᕹᐧ | ᐏᐅᕹᐧ | ᒥᐅᕹᐧ |
| Wife | ᓂᐍ | ᑭᐍ | ᐏᐍ | ᒥᐍ |
| Uncle (paternal) | ᓄᔕᐍᔅ | ᑯᔕᐍᔅ | ᐊᐦᔕᐍᔅ | ᒧᔕᐍᔅ |
| Uncle (maternal) | ᓀᐦᔑᔅ | ᑫᐦᔑᔅ | ᐊᐦᔑᔅ | ᒣᐦᔑᔅ |
| Aunt (paternal) | ᓀᐦᒥᘩᑕᐠᐦᒃ | ᑫᐦᒥᘩᑕᐠᐦᒃ | ᐊᐦᒥᘩᑕᐠᐦᒃ | ᒣᐦᒥᘩᑕᐠᐦᒃ |
| Aunt (maternal) | ᓀᐦᑫᐦᑯᐦᔥ | ᑫᐦᑫᐦᑯᐦᔥ | ᐊᐦᑫᐦᑯᐦᔥ | ᒣᐦᑫᐦᑯᐦᔥ |
| Cousin (male) | ᓀᐦᑕᓐᑯᐧᐦᔅ | ᑫᐦᑕᓐᑯᐧᐦᔅ | ᐊᐦᑕᓐᑯᐧᐦᔅ | ᒣᐦᑕᓐᑯᐧᐦᔅ |
| Cousin (female) | ᓀᐦᑕᓐᑫᐧᐦᓱᖙᐧ | ᑫᐦᑕᓐᑫᐧᐦᓱᖙᐧ | ᐊᐦᑕᓐᑫᐧᐦᓱᖙᐧ | ᒣᐦᑕᓐᑫᐧᐦᓱᖙᐧ |
| Friend (inside) | ᐏᘜᐳ | ᐏᘜᐳ | ᐏᘜᐳ | ᐏᘜᐳ |
| Friend (outside) | ᓂᑕᐠᐦᑉ | ᑭᑕᐠᐦᑉ | ᐅᑕᐠᐦᑉ | ᒥᑕᐠᐦᑉ |
Other Titles (roman)
| English | Powhatan | Script |
| Bachelor | MaTaWoWi | ᒪᑕᐓᐏ |
| Beggar | KuhTaSaMai | ᑫᐦᑕᓴᒪᑊ |
| Chief (district) male | WeRo’Ans | ᐍᘐᐊᓐᔅ |
| Chief (district) female | WeRo’AnSKWa | ᐍᘐᐊᓐᖚᐧ |
| Chief (paramount) | MaMaNaToWik | ᒪᒪᓇᑐᐏᒃ |
| Cuckold | WePenTer | ᐍᐯᓐᑌᖖ |
| Elder (w/ rank or position) | SaKirs | ᓴᑭᖖᔅ |
| Elder w/o rank or position) | NuSa’Anzh | ᓄᐦᓴᐊᓐᒾ |
| Councilor | KeRoNoCho’Es | ᑫᘐᓄᒍᐁᔅ |
# PowerPC Assembly/Instructions
## Instruction Forms

All PowerPC instructions are four bytes long. Whenever the processor calculates the destination address of a branch, the two low-order bits are ignored, so the actual two low-order bits are always 0 in the destination address (i.e., every instruction is word-aligned).
Bits 0-5 of an instruction are specified by the op code, and also indicate the form of the instruction (how to divide up the remaining bits of the instruction into fields, and the meaning of each field).

## Further reading

* ↑ "POWER family and PowerPC Architecture Overview : Instruction Forms".
* "Assembler Language Reference for for POWER-based architectures". (includes both Power2 and PowerPC instruction sets).
# PowerPC Assembly

The PowerPC processor was developed by the AIM Alliance, and the initial specification was released in 1993.
This book is an attempt to teach assembly language programming for the PowerPC processor.

## Table of Contents

* Introduction
* Instructions
# PowerPC Assembly/Introduction
## Why Learn Assembly?

Assembly is the most primitive tool in the programmers toolbox. Entire software projects can be written without ever once looking at a single line of assembly code. So the question arises: why learn assembly? Assembly language is the closest form of communication that humans can engage in with a computer. Using assembly, the programmer can precisely track the flow of data and execution in a program. Also, another benefit to learning assembly, is that once a program has been compiled, it is difficult -- if not impossible -- to decompile the code. That means that if you want to examine a program that is already compiled, you will need to examine it in assembly language. Debuggers also will frequently only show the program code in assembly language. If nothing else, it can be beneficial to learn to read assembly language, if not write it.
Assembly language is also the preferred tool, if not the only tool available for implementing some low-level tasks, such as bootloaders, and low-level kernel components. Code written in assembly has less overhead than code written in high-level languages, so assembly code frequently will run much faster than programs written in other languages. Code that is written in a high-level language can be compiled into assembly, and "hand optimized" to squeeze every last bit of speed out of a section of code. As hardware manufacturers such as Intel and AMD add new features and new instructions to their processors, often times the only way to access those features is to use assembly routines. That is, at least until the major compiler vendors add support for those features.
Developing a program in assembly can be a very time consuming process, however. While it might not be a good idea to write new projects in assembly language, it is certainly valuable to know a little bit about assembly language anyway.

## Who is This Book For?

This book is for those who are learning assembly for the first time, or hoping to optimize their PPC specific code, or even those hoping to learn more about a RISC processor.
# Pinyin/My self-account

My self-account
I do not lack ambition. I do not lack confidence. I do not lack dreams.
The heaven originally creates everything equally. Although I do not live in a good environment, I have a warm and happy family.
woo d zihshuh
woo bu quefar zhihqih. woo bu quefar zihxihn. woo bu quefar mehngxiaang.
shahngtian beenlair jiuh heen gongpirng. suirarn woo bu shenghuor zaih youyueh d huarnjihng, dahn woo yoou 1'g wennuaan xihngfur d jia.
# Pinyin/Narration

Narrative essays use narration as a main method of expression, is a genre with main content that write about the experiences of people and the developments and changes of matters. Narrative essays are used to describe things. Time, people, places, developments, experiences and consequences are the 6 writing elements of narrative essays. For writing objects, need to write about the states of motion, the states of objects or changes. Narrative essays are to let the personal feelings and experiences of self, through the vivid language of imagery, to narrate to readers.
Jìxùwén shì yǐ xùshù wéi zhǔyào biǎodá fāngshì, yǐ xiě rénwù de jīnglì hé shìwù fāzhǎn biànhuà wéi zhǔyào nèiróng de yīzhǒng wéntǐ. Jìxùwén shì yònglái miáoshù shìwù de wénzhāng. Shíjiān, rénwù, dìdiǎn, fāzhǎn, jīngguò, jiéguǒ shì jìxùwén de 6 ge yàosù. Miáoxiě wùtǐ de jiù yào cóng yùndòng zhuàngtài, wùtǐ xíngtài huò biànhuà shàng lái shuō le. Jìxùwén xiězuò, shì bǎ zìjǐ de qīnshēn gǎnshòu hé jīnglì, tōngguò shēngdòng, xíngxiàng de yǔyán, miáoshù gěi dúzhě.
# Pinyin/Nasopharynx cancer

Special:search/nasopharynx cancer prefix:Pinyin/
Special:search/cancer prefix:Pinyin/
Nasopharynx cancer is a malignant tumor of the mucous membrane of nasopharynx, also called "Guangdong tumor". WHO research reported, 80% of nasopharynx cancer sufferers are in China. The incidences of nasopharynx cancer are higher in southern China, such as in the provinces of Guangdong, Guangxi, Hunan, etc. The incidences are even higher, especially in the regions of Zhaoqing, Foshan and Guangzhou in the central and western part of Guangdong province. According to reports, males who live in the central part of Guangdong province and males who speak Cantonese, the rate of morbidity are 30/100 thousand～50/100 thousand. Nationally, the morbidity rates of nasopharynx cancer decrease from south to north, such as the morbidity rates in the most north not higher than 2/100 thousand～3/100 thousand. The age at which nasopharynx cancer occurs from the age of 20 onwards, increasing with age, peaking at 45-60 years old.
Bíyān'ái shì bíyān niānmó de èxìng zhǒngliú, yě jiàozuò “Guǎngdōng zhǒngliú”. Shìjiè wèishēng zǔzhī diàochá bàodào, quánqiú yǒu 80% de bíyān'ái huànzhě zài Zhōngguó. Bíyān'ái de fābìnglǜ yǐ Zhōngguó de nánfāng jiàogāo, rú Guǎngdōng, Guǎngxī, Húnán děng shěng, yóuqíshì Guǎngdōng de zhōngbù hé xībù de Zhàoqìng, Fóshān hé Guǎngzhōu dìqū gèng gāo. Jù bàodào, jūzhù zài Guǎngdōng Shěng zhōngbù yǐjí jiǎng Guǎngdōnghuà de nánxìng, qí fābìnglǜ shì 30/10 wàn ~ 50/10 wàn. Duì quánguó láishuō, bíyān'ái de fābìnglǜ yóu nán dào běi zhújiàn jiàngdī, rú zuì běifāng de fābìnglǜ bù gāoyú 2/10 wàn ~ 3/10 wàn. Bíyān'ái de fābìng niánlíng yóu 20 duō suì kāishǐ, zhújiàn shàngshēng, 45 zhì 60 suì wéi zuì gāofēng.
# Pinyin/Myanmar

Search Pinyin Myanmar
# Pinyin/Name

A name is a word or words that a particular person, animal, place or thing is known by.
mirngzih shih 1'g yiizhi d tehdihng d rern, dohngwuh, dihfang huoh dong'xi d mirngcheng.
# Pinyin/Narrow sense

Search Pinyin sense
# Pinyin/Nasopharynx

See also
# Pinyin/Nanchang

Nanchang is the capital of Jiangxi province of China. It is the political, economical, cultural, commercial, educational, technological and transportation center of Jiangxi. It is one of the core city of the city groups of Chang Jiang river midstream. It is the core city of the ecological economic zone of Poyang lake. It is an important transportation center and a manufacturing base of China. Nanchang is situated in the north of Jiangxi. It is located downstream of Gan Jiang and Fu He rivers. It borders with the southwest of China's biggest freshwater lake Poyang lake. Its area is 7,402.36km2. Its population is 5,200,000+.
Narnchang shih ZG Jiangxi Sheeng d sheenghuih. ta shih Jiangxi d zhehngzhih, jingjih, wernhuah, shangyeh, jiaohyuh, kejih her jiaotong zhongxin. ta shih Charng Jiang Zhongyour Cherngshihqurn d zhongxin cherngshih zhiyi. ta shih Poryarng Hur Shengtaih Jingjihqu d herxin cherngshih. ta shih ZG zhohngyaoh d jiaotong zhongxin her zhihzaohyeh jidih. Narnchang weihyur Jiangxi d beeibuh, Gahn Jiang her Fuu Her d xiahyour. ta yuu ZG zuihdah dahnshuiihur Poryarng Hur d xinarnbuh xiangliarn. ta d miahnji 7402.36km2, rernkoou 520'0000+.
# Pinyin/Tomorrow

See also
Tomorrow, is the beginning of winning of the strong; tomorrow, is the putting off by the laziness of the weak; tomorrow, is the newborn splendour of the wise; tomorrow, is the often used lies of the stupid; facing tomorrow, only those without remorse of time, can calmly; face tomorrow, only those who know life values, will not hide; facing tomorrow, with no accomprishment, is the behaviour of coward; facing tomorrow, should be able to do no remorse no hatred, carrying forward!
Míngtiān, shì qiángzhě shènglì de kāiduān; míngtiān, shì ruòzhě lǎnduò de fūyǎn; míngtiān, shì zhìzhě xīnshēng de cuǐcàn; míngtiān, shì yúzhě guànyòng de huǎngyán; miànduì míngtiān, zhǐyǒu wúkuì yú shíjiān de rén, cái huì tǎnrán; miànduì míngtiān, zhǐyǒu dǒngdé shēngmìng jiàzhí de rén, cái bùhuì duǒshǎn; miànduì míngtiān, lùlùwúwéi, shì nuòfū de biǎoxiàn; miànduì míngtiān, yīnggāi zuòdào wúhuǐwúyuàn, yǒngwǎngzhíqián!
# Pinyin/To observe

To observe
You stand on the bridge to view scenery, the one who look at scenery look at you from above at the building. Bright moon decorates your window, you decorate the dreams of others.
Duànzhāng
Nǐ zhàn zài qiáo shàng kàn fēngjǐng, kàn fēngjǐng de rén zài lóushàng kàn nǐ. Míngyuè zhuāngshìle nǐ de chuāngzi, nǐ zhuāngshìle biérén de mèng.
# Pinyin/Tobacco

See also
Tobacco products are products made entirely or partly of leaf tobacco as raw material, which are intended to be smoked, sucked, chewed or snuffed. All contain the highly addictive psychoactive ingredient, nicotine.
Tobacco use is one of the main risk factors for a number of chronic diseases, including cancer, lung diseases, and cardiovascular diseases. Despite this, it is common throughout the world. A number of countries have legislation restricting tobacco advertising, and regulating who can buy and use tobacco products, and where people can smoke.
yancaao zhìpǐn shì wánquán huò bùfèn yòng yānyè zuòwéi yuánliào zhìchéng de chǎnpǐn, yòngyú chōuxī, jǔjué huò bíxī. Suǒyǒu bāohán nicotine zhèzhǒng fēicháng ràng rén shàngyǐn érqiě néng zuòyòng yú jīngshén de chéngfèn.
Yāncǎo shǐyòng shì dǎozhì yīxìliè mànxìngbìng, bāokuò áizhèng, fèibìng hé xīnxuèguǎnbìng de zhǔyào wēixiǎn yīnsù zhīyī. Jǐnguǎn rúcǐ, yāncǎo shǐyòng zài shìjiè gèdì dōu hěn chángjiàn. Yīxiē guójiā yǐjīng yǒu fǎlǜ xiànzhì yāncǎo guǎnggào bìngqiě duì gòumǎi hé shǐyòng yāncǎo zhìpǐn de rén, yǐjí yǔnxǔ xīyān de chǎngsuǒ jìnxíng guǎnzhì.
# Pinyin/Together

See also
# Pinyin/Tonight

See alsotonight
# Pinyin/Tone

See also
A tone is the degree to which the sound of a syllable is high or low.
shengdiaoh shih yinjier fayin gaodi d cherngduh.
# Pinyin/Tomato

Search Pinyin tomato
Tomato is a red fruit. It is shiny and smooth, and has many small seeds. The tomato is green when it is unripe. It is very sour and it is good for you. It slowly changes color from green to red as it gets more ripe. There are many different types of tomatoes. Most tomatoes are red, but some special kinds are yellow or orange when they are ripe. Also, some tomatoes are as small as strawberries, and some will become as big as apples. Tomatoes are used a lot in Italian food. They are also used to make ketchup. Sometimes, people mistake tomatoes for vegetables, but they aren't, because they contain seeds. Tomato seeds are dispersed by being eaten by, and then passing through, animals.
Fānqié shì hóngsè shuǐguǒ, yǒu guāngzé, guānghuá, bìng yǒu xǔduō xiǎo zhǒngzǐ. Fānqié wèi chéngshú shí lǜsè, hěn suān, dàn duì shēntǐ hǎo. Tā zhújiàn chéngshú, cóng lǜsè biàn hóngsè. Fānqié yǒu hěnduō bùtóng lèixíng, chéngshú shí dàduō hóngsè, yīxiē tèshū pǐnzhǒng huángsè huò chéngsè. Yǒuxiē fānqié xiǎo rú cǎoméi, yǒude dà rú píngguǒ. Fānqié chángyòng yú Italy shípǐn, yě yòngyú zào fānqiéjiàng. Yǒurén yǐwéi fānqié shì shūcài, qíshí shì shuǐguǒ, yīnwèi tā yǒu zhǒngzǐ. Dòngwù chī fānqié hòu páichū zhǒngzǐ, cóng'ér bāngzhù bōzhòng.
# Ordinary Differential Equations/Higher Order

Differential equations of the second order and higher are generally much more difficult than first-order DEs to solve, and require new methods to solve them. First we will look at how we have to treat higher-order DEs differently, then we will move on to actually solving them.

### 2: Homogeneous Second-Order Equations

* Explanation
* Examples
* Problems
* Answers

### 3: Non-Homogeneous Second-Order Equations

* Explanation
* Examples
* Problems
* Answers
# Ordinary Differential Equations/Homogenous 3

1) $3y''+18y'-81y=0$
2) $y''+6y'+13y=0$
3) $y''+10y'+25y=0$
4) $y''''+24y'''+218y''+838y'+1369y=0$
5) $y'''-2y''-15y'+36y=0$
6) $y'''+5y''-4y'-20y=0$
7) $y'''+4y''+y'-26y=0$
# Ordinary Differential Equations/Laplace Transform
## Definition

Let $f(t)$ be a function on $[0,\infty )$. The Laplace transform of $f$ is defined by the integral
    $F(s)={\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }e^{-st}f(t)dt\,.$
The domain of $F(s)$ is all values of $s$ such that the integral exists.

## Properties
### Linearity

Let $f$ and $g$ be functions whose Laplace transforms exist for $s>\alpha$ and let $a$ and $b$ be constants. Then, for $s>\alpha$,
    ${\mathcal {L}}\{af+bg\}=a{\mathcal {L}}\{f\}+b{\mathcal {L}}\{g\}\,,$
which can be proved using the properties of improper integrals.

### Shifting in s

If the Laplace transform ${\mathcal {L}}\{f\}(s)=F(s)$ exists for $s>\alpha$, then
    ${\mathcal {L}}\{e^{at}f(t)\}(s)=F(s-a)\,$
for $s>\alpha +a$.
Proof.
${\begin{aligned}{\mathcal {L}}\{e^{at}f(t)\}(s)&=\int _{0}^{\infty }e^{-st}e^{at}f(t)dt\\&=\int _{0}^{\infty }e^{-(s-a)t}f(t)dt\\&=F(s-a)\,.\end{aligned}}$

### Laplace Transform of Higher-Order Derivatives

If $F(s)={\mathcal {L}}\{f(t)\}$, then ${\mathcal {L}}\{f'(t)\}=sF(s)-f(0)$
    Proof:
       ${\mathcal {L}}\{f'(t)\}=\int _{0}^{\infty }f'(t)e^{-st}dt$
       $=\lim _{C\to \infty }\int _{0}^{C}f'(t)e^{-st}dt$
       $=\lim _{C\to \infty }\left.e^{-st}f(t)\right|_{0}^{C}-\int _{0}^{C}-sf(t)e^{-st}dt$ (integrating by parts)
       $=-f(0)+s\lim _{C\to \infty }\int _{0}^{C}f(t)e^{-st}dt$
       $=s{\mathcal {L}}\{f(t)\}-f(0)$
       $=sF(s)-f(0)\,$
Using the above and the linearity of Laplace Transforms, it is easy to prove that ${\mathcal {L}}\{f''(t)\}=s^{2}F(s)-sf(0)-f'(0)$

### Derivatives of the Laplace Transform

If ${\mathcal {L}}\{f(t)\}=F(s)$, then ${\mathcal {L}}\{tf(t)\}=-F'(s)$

## Laplace Transform of Few Simple Functions

* ${\mathcal {L}}\{1\}={1 \over s}$
* ${\mathcal {L}}\{e^{at}\}={1 \over s-a}$
* ${\mathcal {L}}\{\cos \omega t\}={s \over s^{2}+\omega ^{2}}$
* ${\mathcal {L}}\{\sin \omega t\}={\omega \over s^{2}+\omega ^{2}}$
* ${\mathcal {L}}\{1\}={1 \over s}$
* ${\mathcal {L}}\{t^{n}\}={n! \over s^{n+1}}$