Sign In Sign Up Ask an AI advisor

rah45

rah45

27.07.2019 •

Mathematics

Find the general solution of yʹ − 3y = 8e3t + 4sin t .

Solved

Show answers

Ответ:

mcalepcrager

14.03.2022

$ye^{-3t} = 8t - \frac{2e^{-3t}}{5}(3sin t + cost)+C'$

Step-by-step explanation:

Given differential equation,

$y' - 3y = 8e^{3t} + 4sin t$

$\frac{dy}{dt}-3y = 8e^{3t} + 4sin t$

Since, the above equation is of the type of linear differential equation

$\frac{dy}{dx}+Py=Q$ ,

In which P = -3, Q = $8e^{3t} + 4sin t$ ,

Thus, the integrating factor,

$I.F. = e^{\int (-3) dt}=e^{-3t}$

Hence, the solution of the given differential equation would be,

$y\times I.F. = \int I.F.\times Q dt$

$\implies y\times e^{-3t}=\int e^{-3t}\times (8e^{3t} + 4sin t)=\int 8+4e^{-3t} sin tdt$

$y\times e^{-3t} = \int 8 dt + 4\int e^{-3t} sin tdt$

$\implies y\times e^{-3t} = 8t + 4\int e^{-3t} sin tdt------(1)$

Let,

$I=\int e^{-3t} sin tdt-----(2)$

Integrating by parts, ( First term = sin t, second term = $e^{-3t}$ )

$I=-\frac{e^{-3t} sint}{3} - \int -\frac{e^{3t} cost}{3} dt+C$

$I=-\frac{e^{-3t} sint}{3} - [\frac{e^{-3t} cost }{9}-\int -\frac{e^{-3t} sint}{9}dt]+C$

$I=-\frac{e^{-3t} sint}{3} - [\frac{e^{-3t} cost }{9}+\frac{1}{9}\int e^{-3t} sint dt]+C$

$I=-\frac{e^{-3t} sint}{3} - [\frac{e^{-3t} cost }{9}+\frac{1}{9} I]+C$ ( from equation (2))

$I+\frac{I}{9}=-\frac{e^{-3t} sint}{3} - \frac{e^{-3t} cost }{9}+C$

$\frac{10}{9}I=\frac{-3e^{-3t} sint-e^{-3t} cost }{9}+C$

$I=\frac{1}{10}(-3e^{-3t} sint-e^{-3t} cost)+C'$ ( where, C' = 9C/10 )

$I=-\frac{e^{-3t}}{10}(3sin t + cost)+C'$

From equation (1),

$y\times e^{-3t} = 8t - \frac{4e^{-3t}}{10}(3sin t + cost)+C'$

$ye^{-3t} = 8t - \frac{2e^{-3t}}{5}(3sin t + cost)+C'$

0,0(0 оценок)

Ответ:

BobJRJR

20.06.2020

sides are proportional and angles are congruent

Step-by-step explanation:

Every segment in ABCDEFG is parallel to, and double the length of, the corresponding segment in HIJKLMN. All corresponding angles are congruent. Thus, the definition of similar polygons is met:

corresponding sides are angles are congruent.

0,0(0 оценок)

Ask your question to the AI advisor

Ai-bot is an expert in any field and is the perfect companion for reliable and useful answers and advice on a variety of topics, including science, history, technology, art, sports, health, culture and more.

Ask your question

More tips

C Computers and Internet What is a URL? Everything You Need to Know About URL Addresses...
C Computers and Internet Why is Yandex s robot Bless.yandex.ru Important?...
F Food and Cooking Fresh and Easy Lemonade in 10 Minutes...
S Sport How to wrap boxing hand wraps? Everything you need to know!...
F Family and Home How to Sew Curtain Tapes: Best Tips from Professionals...
A Animals and plants How to Grow Lime from a Seed: Simple Tips and Interesting Facts...
C Computers and Internet How to Create a Folder on Your iPhone?...
G Goods and services How to sew a ribbon: Tips for beginners...
F Food and Cooking How to Make Mayonnaise at Home? Secrets of Homemade Mayonnaise...
C Computers and Internet Which Phone is Best for Internet Surfing?...

Answers on questions: Mathematics

M Mathematics The following table summarizes a students favorite season and whether or not the student has allergies. Has Allergies No Allergies Total Winter 5 9 14 Spring 6 7 13 Summer 13 23...
M Mathematics Ineed plz : ) what type of angle is shown? straight acute obtuse right...
M Mathematics Bored out my brain and ideas to get unbored?...
H Health Which of the following is incorrect concerning school-age children s perceptual development? (a) school-age children do more in-depth exploring of their world, using their eyes...
B Biology What name is given to genetically identical organisms?...
M Mathematics Since the area of the circle is I the area of the square, the volume of the cone equals o I the volume of the pyramid or (ench or srh. o the volume of the pyramid or (2n o the...
E English He subject of the lecture will affect the way a student takes notes. select the best answer from the choices provided t f...

S Sport S Society and Politics G Goods and services C Cities and Countries A Animals and plants L Legal consultation W Work and Career C Construction and repair H Health and Medicine D Dating, Love, Relationships H Horoscopes, Magic, Divination S Style and Beauty L Leisure and Entertainment S Science and Technology P Photography and Videography A Art and Culture O Other P Philosophy F Food and Cooking B Business and Finance T Travel and tourism F Family and Home A Auto and Moto

Show more

Full access

It will provide effective and fast learning by providing unlimited access to the knowledge base and receiving answers from experts and an AI advisor.

Subscribe

Ask an AI advisor a question

I want advice

ABOUT US

About us
Terms of use
Copyright
Privacy Policy
Cookie policy
Cookie preferences

HELP

Sign Up
FAQ
Security Center

App

Spanish Geography Health Engineering Arts Social Studies Advanced Placement (AP) German SAT History Computers and Technology Physics Chemistry Biology Law Mathematics World Languages French Business Medicine