{
localUrl: '../page/first_order_linear_equation.html',
arbitalUrl: 'https://arbital.com/p/first_order_linear_equation',
rawJsonUrl: '../raw/845.json',
likeableId: '4023',
likeableType: 'page',
myLikeValue: '0',
likeCount: '1',
dislikeCount: '0',
likeScore: '1',
individualLikes: [
'FaisalAlZaben'
],
pageId: 'first_order_linear_equation',
edit: '2',
editSummary: '',
prevEdit: '1',
currentEdit: '2',
wasPublished: 'true',
type: 'wiki',
title: 'First order linear equations',
clickbait: '',
textLength: '2060',
alias: 'first_order_linear_equation',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'JaimeSevillaMolina',
editCreatedAt: '2017-03-28 14:58:58',
pageCreatorId: 'JaimeSevillaMolina',
pageCreatedAt: '2017-03-28 14:50:50',
seeDomainId: '0',
editDomainId: 'arbital_featured_project',
submitToDomainId: '0',
isAutosave: 'false',
isSnapshot: 'false',
isLiveEdit: 'true',
isMinorEdit: 'false',
indirectTeacher: 'false',
todoCount: '0',
isEditorComment: 'false',
isApprovedComment: 'false',
isResolved: 'false',
snapshotText: '',
anchorContext: '',
anchorText: '',
anchorOffset: '0',
mergedInto: '',
isDeleted: 'false',
viewCount: '45',
text: 'A **first order lineal equation** has the form\n$$\nu'=a(t)u+b(t)\n$$\nwhere $a$ and $b$ are continuous functionsfrom an interval $[\\alpha, \\beta]$ to the real line.\n\n$b$ is called the inhomogeneity of the problem, and the equation where $b=0$ is called the associated homogeneous equation.\n$$\nu'=a(t)u\n$$\n\nA **solution** of a first order linear equation is a $C^1$ function from $[\\alpha, \\beta]$ to the real line such that the equation is satisfied at all times. We will denote the set of solutions of an equation with inhomogeneity $b$ as $\\Sigma_b$, and the solutions of the associated homogeneous system as $\\Sigma_0$.\n\n## Properties of the space of solutions\n$\\Sigma_0$ is a [3w0 vector space]; that is, it satifies the **principle of superposition**: linear combinations of solutions are solutions.\n\n$\\Sigma_b$ is an [-affine_space] parallel to $\\Sigma_0$. That is, it satifies that the difference of any two solutions are in $\\Sigma_0$, and any element in $\\Sigma_0$ plus other element in $\\Sigma_b$ is an element from $\\Sigma_b$.\n\n## First order linear equations of constant coefficients\nOne special kind of linear equations are those in which the coefficients $a$ and $b$ are constant numbers Such linear equations are always resoluble.\n$$\nu' = au+b\n$$\n\nTo solve them, we first have to solve the associated homogeneous equation $u'=au$.\n\nThis has as a solution the functions $ke^{\\int_{t_0}^ta}$ for $k$ constant and $t_0\\in [\\alpha, \\beta]$.\n\nWe can find a concrete solution of the inhomogeneous equation using **variation of coefficients**.\nWe consider as a candidate to a solution the function $u=h\\dot v$, for $h$ a solution of the homogeneous system such as $e^{\\int_{t_0}^ta}$.\n\nThen if we plug $u$ into the equation we find that\n$$\nu'=(hv)'=h'v+hv'=au+b=a(hv)+b\n$$\nSince $h\\in\\Sigma_0$, $h'=ah$, thus\n$$\nv'=bh^{-1}=be^{-\\int_{t_0}^ta}\n$$\nTherefore we can integrate and we arrive to:\n$$\nv=\\int_{t_0}^tbe^{\\int_{t}^sa}ds\n$$\nBy the affinity of $\\Sigma_b$, we can parametrize it by $ke^{\\int_{t_0}^ta}+\\int_{t_0}^tbe^{\\int_{t}^sa}ds$ for $k$ constant.',
metaText: '',
isTextLoaded: 'true',
isSubscribedToDiscussion: 'false',
isSubscribedToUser: 'false',
isSubscribedAsMaintainer: 'false',
discussionSubscriberCount: '2',
maintainerCount: '1',
userSubscriberCount: '0',
lastVisit: '',
hasDraft: 'false',
votes: [],
voteSummary: [
'0',
'0',
'0',
'0',
'0',
'0',
'0',
'0',
'0',
'0'
],
muVoteSummary: '0',
voteScaling: '0',
currentUserVote: '-2',
voteCount: '0',
lockedVoteType: '',
maxEditEver: '0',
redLinkCount: '0',
lockedBy: '',
lockedUntil: '',
nextPageId: '',
prevPageId: '',
usedAsMastery: 'false',
proposalEditNum: '0',
permissions: {
edit: {
has: 'false',
reason: 'You don't have domain permission to edit this page'
},
proposeEdit: {
has: 'true',
reason: ''
},
delete: {
has: 'false',
reason: 'You don't have domain permission to delete this page'
},
comment: {
has: 'false',
reason: 'You can't comment in this domain because you are not a member'
},
proposeComment: {
has: 'true',
reason: ''
}
},
summaries: {
Summary: 'A **first order lineal equation** has the form'
},
creatorIds: [
'JaimeSevillaMolina'
],
childIds: [],
parentIds: [],
commentIds: [
'84z'
],
questionIds: [],
tagIds: [],
relatedIds: [],
markIds: [],
explanations: [
{
id: '7598',
parentId: 'first_order_linear_equation',
childId: 'first_order_linear_equation',
type: 'subject',
creatorId: 'JaimeSevillaMolina',
createdAt: '2017-03-28 14:50:50',
level: '2',
isStrong: 'true',
everPublished: 'true'
}
],
learnMore: [],
requirements: [],
subjects: [
{
id: '7598',
parentId: 'first_order_linear_equation',
childId: 'first_order_linear_equation',
type: 'subject',
creatorId: 'JaimeSevillaMolina',
createdAt: '2017-03-28 14:50:50',
level: '2',
isStrong: 'true',
everPublished: 'true'
}
],
lenses: [],
lensParentId: '',
pathPages: [],
learnMoreTaughtMap: {},
learnMoreCoveredMap: {},
learnMoreRequiredMap: {},
editHistory: {},
domainSubmissions: {},
answers: [],
answerCount: '0',
commentCount: '0',
newCommentCount: '0',
linkedMarkCount: '0',
changeLogs: [
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '22408',
pageId: 'first_order_linear_equation',
userId: 'JaimeSevillaMolina',
edit: '2',
type: 'newEdit',
createdAt: '2017-03-28 14:58:58',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '22405',
pageId: 'first_order_linear_equation',
userId: 'JaimeSevillaMolina',
edit: '1',
type: 'newEdit',
createdAt: '2017-03-28 14:50:50',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '22406',
pageId: 'first_order_linear_equation',
userId: 'JaimeSevillaMolina',
edit: '0',
type: 'newTeacher',
createdAt: '2017-03-28 14:50:50',
auxPageId: 'first_order_linear_equation',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '22407',
pageId: 'first_order_linear_equation',
userId: 'JaimeSevillaMolina',
edit: '0',
type: 'newSubject',
createdAt: '2017-03-28 14:50:50',
auxPageId: 'first_order_linear_equation',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'false',
hasParents: 'false',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}