SOLUTION: Direct Integration Separation of Variables & Integrating Factor Method Questions

Hi. I have submitted your work. It is in doc format due to Studypool’s quality requirements. If there is any room for improvement, please do let me know. Have a great day. Section 1.1:Answer 1:Given:𝑦 β€² = 3π‘₯ 2 _______(1)𝑦 = π‘₯ 3 + 7 _______(2)Here we need to prove the solution of a given differential equation using substitution.Equation (1) can also be written as:𝑑𝑦= 3π‘₯ 2𝑑π‘₯Substitute Equation (2) to Equation (1), we get:𝑑(π‘₯ 3 + 7)= 3π‘₯ 2𝑑π‘₯Solving the L.H.S further:𝑑(π‘₯ 3 ) 𝑑(7)+= 3π‘₯ 2𝑑π‘₯𝑑π‘₯3π‘₯ 2 = 3π‘₯ 2Therefore L.H.S. = R.H.S.Hence Proved.Answer 19:Given:𝑦 β€² = 𝑦 + 1 _______(1)𝑦(π‘₯) = 𝐢𝑒 π‘₯ βˆ’ 1 _______(2)𝑦(0) = 5 _______(3)Calculating the derivative of Equation (2):𝑦 β€² (π‘₯) = 𝐢𝑒 π‘₯ _______(4)Substituting Equation (2) into Equation (1):𝑦 β€² = 𝐢𝑒 π‘₯ βˆ’ 1 + 1𝑦 β€² = 𝐢𝑒 π‘₯This returns the value of Equation (4) which was the derivative of Equation (2)Now Solving for the value of C, substitute value of 𝑦(0) to Equation (2):𝑦(0) = 𝐢𝑒 0 βˆ’ 15=πΆβˆ’1𝐢 =5+1𝐢=6The given initial condition is:π’š(𝒙) = πŸ”π’†π’™ βˆ’ 𝟏Section 1.2:Answer 2:Given:𝑑𝑦= (π‘₯ βˆ’ 2)2𝑑π‘₯𝑦(2) = 1In order to find the required equation w.r.t the given initial condition, we need to take the antiderivative of the above differential equation:𝑑𝑦= (π‘₯ βˆ’ 2)2𝑑π‘₯𝑑𝑦 = (π‘₯ βˆ’ 2)2 𝑑π‘₯∫ 𝑑𝑦 = ∫(π‘₯ βˆ’ 2)2 𝑑π‘₯𝑦=(π‘₯ βˆ’ 2)3+𝐢3The value of C is unknown. Plugging in the value of 𝑦(2) = 1 to find the value of C.(2 βˆ’ 2)3𝑦(2) =+𝐢31= 0+𝐢𝐢=1The solution is:(𝒙 βˆ’ 𝟐)πŸ‘π’š=+πŸπŸ‘Answer 13:Given:π‘Ž(𝑑) = 3𝑑𝑣0 = 5π‘₯0 = 0In order to find the velocity function, we will integrate the acceleration function w.r.t to β€˜t’:∫ π‘Ž(𝑑) = ∫ 3π‘‘βˆ«π‘‘π‘£= ∫ 3π‘‘π‘‘π‘‘βˆ« 𝑑𝑣 = 3. ∫ 𝑑 𝑑𝑑𝑣=3𝑑 2+ 𝐢12Plugging in the value of initial condition 𝑣(0) = 5 in above equation. Note that v0 stands forvelocity at initial time (t=0)3(0)25=+ 𝐢12𝐢1 = 5Therefore, the velocity function is:πŸ‘π’•πŸπ’—(𝒕) =+πŸ“πŸTaking the anti-derivative again to find the displacement function:βˆ«π‘‘π‘₯3𝑑 2= ∫+5𝑑𝑑23𝑑 2∫ 𝑑π‘₯ = ∫ (+ 5) 𝑑𝑑23 𝑑3π‘₯ = . + 5𝑑 + 𝐢22 3𝑑3π‘₯ = + 5𝑑 + 𝐢22Plugging in the value of π‘₯0 = 0 in above equation to find the initial value equation0=(0)3+ 5(0) + 𝐢22𝐢2 = 0Therefore:𝒙(𝒕) =π’•πŸ‘+ πŸ“π’•πŸAnswer 28:Given:Initial velocity = 𝑣0 = 40 𝑓𝑑/𝑠Height of Monument = 𝑦(𝑑) = 555 π‘“π‘‘π‘šπ‘“π‘‘Acceleration = π‘Ž = 9.81 𝑠2 = 32 𝑠2We know that the second equation of motion is:1βˆ†π‘₯ = 𝑣0 𝑑 + π‘Žπ‘‘ 2 ______(1)2Plugging in the values in above equation;1555 = (40)𝑑 + (32)𝑑 2216𝑑 2 + 40𝑑 βˆ’ 555 = 0Taking the roots of above equation:𝑑=βˆ’40 Β± √402 + 4π‘₯16π‘₯5552π‘₯16π‘œπ‘Ÿ; 𝑑 = 4.78 𝑠𝑒𝑐Performing the derivation on the equation (1) w.r.t β€˜t’ to obtain velocity function:𝑣(𝑑) = 𝑣0 + π‘Žπ‘‘Plugging in the values:𝑣(𝑑) = 40 + 32(4.77)𝒗(𝒕) = πŸπŸ—πŸ. πŸ”πŸ– 𝒇𝒕/𝒔Hence this will be the final velocity of the ball at impact.Section 1.4:Answer 2:Given:𝑑𝑦+ 2π‘₯𝑦 2 = 0𝑑π‘₯In order to find the general solution, it is important to separate the two variables x and y andthen perform integration:𝑑𝑦= βˆ’2π‘₯𝑦 2𝑑π‘₯𝑑𝑦 = βˆ’2π‘₯𝑦 2 𝑑π‘₯1𝑑𝑦 = βˆ’2π‘₯ 𝑑π‘₯𝑦2∫1𝑑𝑦 = βˆ’2 ∫ π‘₯ 𝑑π‘₯𝑦2∫ 𝑦 βˆ’2 𝑑𝑦 = βˆ’2 ∫ π‘₯ 𝑑π‘₯𝑦 βˆ’1 βˆ’2π‘₯ 2=+πΆβˆ’12βˆ’1= βˆ’π‘₯ 2 + 𝐢𝑦1= π‘₯2 βˆ’ πΆπ‘¦π’š=π’™πŸπŸβˆ’π‘ͺAnswer 11:Given:𝑦 β€² = π‘₯𝑦 3π‘œπ‘Ÿ,𝑑𝑦= π‘₯𝑦 3𝑑π‘₯In order to find the general solution, we will separate the two variables x and y and performtheir integration separately:𝑑𝑦 = π‘₯𝑦 3 𝑑π‘₯1𝑑𝑦 = …

Order a unique copy of this paper
(550 words)

Approximate price: $22

Our Basic features
  • Free title page and bibliography
  • Plagiarism-free guarantee
  • Unlimited revisions
  • Money-back guarantee
  • 24/7 support
Our Options
  • Writer’s samples
  • Expert Proofreading
  • Overnight delivery
  • Part-by-part delivery
  • Copies of used sources
Paper format
  • 275 words per page
  • 12 pt Arial/Times New Roman
  • Double line spacing
  • Any citation style (APA, MLA, Chicago/Turabian, Harvard)

AcademicWritingCompany guarantees

Our customer is the center of what we do and thus we offer 100% original essays..
By ordering our essays, you are guaranteed the best quality through our qualified experts.All your information and everything that you do on our website is kept completely confidential.

Money-back guarantee always strives to give you the best of its services. As a custom essay writing service, we are 100% sure of our services. That is why we ensure that our guarantee of money-back stands, always

Read more

Zero-plagiarism tolerance guarantee

The paper that you order at is 100% original. We ensure that regardless of the position you are, be it with urgent deadlines or hard essays, we give you a paper that is free of plagiarism. We even check our orders with the most advanced anti-plagiarism software in the industry.

Read more

Free-revision guarantee

The thrives on excellence and thus we help ensure the Customer’s total satisfaction with the completed Order.To do so, we provide a Free Revision policy as a courtesy service. To receive free revision the Academic writing Company requires that the you provide the request within Fifteen (14) days since the completion date and within a period of thirty (30) days for dissertations and research papers.

Read more

Privacy and Security policy

With, your privacy is the most important aspect. First, the academic writing company will never resell your personal information, which include credit cards, to any third party. Not even your lecturer on institution will know that you bought an essay from our academic writing company.

Read more

Adherence to requirements guarantee

The academic writing company writers know that following essay instructions is the most important part of academic writing. The expert writers will, therefore, work extra hard to ensure that they cooperate with all the requirements without fail. We also count on you to help us provide a better academic paper.

Read more

Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2020 at 10:52 AM
Total price:
The price is based on these factors:
Customer Academic level
Number of pages required
Urgency of paper