Nos Problemas 1-12, ache a solução geral do sistema dado. X' = (-1 -1 0; 3/4 -3/2 3; 1/8 1/4 -1/2) X

Passo 1

Vamos começar entendendo o que o problema está pedindo. Temos um sistema de equações diferenciais lineares da forma , onde A é a matriz dada:
Nosso objetivo é encontrar a solução geral desse sistema.

Passo 2

Para resolver esse sistema, vamos utilizar o método dos autovalores e autovetores. Isso envolve encontrar os autovalores ( lambda) da matriz A e, para cada autovalor, encontrar os correspondentes autovetores ( v). A solução geral será uma combinação linear dessas soluções básicas.

Passo 3

Primeiro, vamos encontrar os autovalores de A. Para isso, precisamos resolver a equação característica:
I
Onde é a matriz identidade de ordem 3. Vamos calcular lambda:
Agora, vamos calcular o determinante dessa matriz:
Simplificando os cálculos:
Continuando o processamento, obtemos os valores de que satisfazem a equação característica.

Passo 4

Após resolver a equação característica, encontramos os autovalores , , e . Para cada autovalor, precisamos encontrar o autovetor correspondente, resolvendo o sistema .
Por exemplo, para um autovalor , resolvemos:

[


]
Resolvendo esse sistema, encontramos os componentes do autovetor correspondente.

Passo 5

Repetimos o processo para os demais autovalores e , encontrando os respectivos autovetores e .

Resposta

A solução geral do sistema é dada por:
Onde , e são constantes determinadas pelas condições iniciais, são os autovalores encontrados, e são os respectivos autovetores.

Exercícios relacionados

<html><head></head><body><div>Nos Problemas 1-12, ache a solução geral do sistema dado. <span class="math-inline ">
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<html><head></head><body><div>Nos Problemas 1-12, ache a solução geral do sistema dado. <span class="math-inline ">
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http://www.w3.org/2000/svg” width=“17.192ex” height=“5.43ex” role=“img” focusable=“false” viewBox=“0 -1450 7599 2400”><title>X’ = (-6 2; -3 1) X</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“msup”><g data-mml-node=“TeXAtom” data-mjx-texclass=“ORD”><g data-mml-node=“mi”><path data-c=“1D417” d=“M327 0Q306 3 174 3Q52 3 43 0H33V62H98L162 63L360 333L157 624H48V686H59Q80 683 217 683Q368 683 395 686H408V624H335L393 540L452 458L573 623Q573 624 528 624H483V686H494Q515 683 646 683Q769 683 778 686H787V624H658L575 511Q493 398 493 397L508 376Q522 356 553 312T611 229L727 62H835V0H824Q803 3 667 3Q516 3 489 0H476V62H513L549 63L401 274L247 63Q247 62 292 62H338V0H327Z”></path></g></g><g data-mml-node=“TeXAtom” transform=“translate(902,363) scale(0.707)” data-mjx-texclass=“ORD”><g data-mml-node=“mi”><path data-c=“2032” d=“M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 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<html><head></head><body><div>Nos Problemas 1-12, ache a solução geral do sistema dado. <span class="math-inline "></span></div>
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<html><head></head><body><div>Encontre a primitiva mais geral de cada uma das seguintes funções:<br>
<span class=“math-inline “>
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.566ex;” xmlns=“
http://www.w3.org/2000/svg” width=“20.594ex” height=“2.681ex” role=“img” focusable=“false” viewBox=“0 -935 9102.7 1185”><title>f(t) = 2 * raiz(t) + 6 * cos(t)</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“1D453” d=“M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z”></path></g><g 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