# 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 ′ = 3𝑥 2 _______(1)𝑦 = 𝑥 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= 3𝑥 2𝑑𝑥Substitute Equation (2) to Equation (1), we get:𝑑(𝑥 3 + 7)= 3𝑥 2𝑑𝑥Solving the L.H.S further:𝑑(𝑥 3 ) 𝑑(7)+= 3𝑥 2𝑑𝑥𝑑𝑥3𝑥 2 = 3𝑥 2Therefore L.H.S. = R.H.S.Hence Proved.Answer 19:Given:𝑦 ′ = 𝑦 + 1 _______(1)𝑦(𝑥) = 𝐶𝑒 𝑥 − 1 _______(2)𝑦(0) = 5 _______(3)Calculating the derivative of Equation (2):𝑦 ′ (𝑥) = 𝐶𝑒 𝑥 _______(4)Substituting Equation (2) into Equation (1):𝑦 ′ = 𝐶𝑒 𝑥 − 1 + 1𝑦 ′ = 𝐶𝑒 𝑥This 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) = 𝐶𝑒 0 − 15=𝐶−1𝐶 =5+1𝐶=6The given initial condition is:𝒚(𝒙) = 𝟔𝒆𝒙 − 𝟏Section 1.2:Answer 2:Given:𝑑𝑦= (𝑥 − 2)2𝑑𝑥𝑦(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= (𝑥 − 2)2𝑑𝑥𝑑𝑦 = (𝑥 − 2)2 𝑑𝑥∫ 𝑑𝑦 = ∫(𝑥 − 2)2 𝑑𝑥𝑦=(𝑥 − 2)3+𝐶3The value of C is unknown. Plugging in the value of 𝑦(2) = 1 to find the value of C.(2 − 2)3𝑦(2) =+𝐶31= 0+𝐶𝐶=1The solution is:(𝒙 − 𝟐)𝟑𝒚=+𝟏𝟑Answer 13:Given:𝑎(𝑡) = 3𝑡𝑣0 = 5𝑥0 = 0In order to find the velocity function, we will integrate the acceleration function w.r.t to ‘t’:∫ 𝑎(𝑡) = ∫ 3𝑡∫𝑑𝑣= ∫ 3𝑡𝑑𝑡∫ 𝑑𝑣 = 3. ∫ 𝑡 𝑑𝑡𝑣=3𝑡 2+ 𝐶12Plugging 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=+ 𝐶12𝐶1 = 5Therefore, the velocity function is:𝟑𝒕𝟐𝒗(𝒕) =+𝟓𝟐Taking the anti-derivative again to find the displacement function:∫𝑑𝑥3𝑡 2= ∫+5𝑑𝑡23𝑡 2∫ 𝑑𝑥 = ∫ (+ 5) 𝑑𝑡23 𝑡3𝑥 = . + 5𝑡 + 𝐶22 3𝑡3𝑥 = + 5𝑡 + 𝐶22Plugging in the value of 𝑥0 = 0 in above equation to find the initial value equation0=(0)3+ 5(0) + 𝐶22𝐶2 = 0Therefore:𝒙(𝒕) =𝒕𝟑+ 𝟓𝒕𝟐Answer 28:Given:Initial velocity = 𝑣0 = 40 𝑓𝑡/𝑠Height of Monument = 𝑦(𝑡) = 555 𝑓𝑡𝑚𝑓𝑡Acceleration = 𝑎 = 9.81 𝑠2 = 32 𝑠2We know that the second equation of motion is:1∆𝑥 = 𝑣0 𝑡 + 𝑎𝑡 2 ______(1)2Plugging in the values in above equation;1555 = (40)𝑡 + (32)𝑡 2216𝑡 2 + 40𝑡 − 555 = 0Taking the roots of above equation:𝑡=−40 ± √402 + 4𝑥16𝑥5552𝑥16𝑜𝑟; 𝑡 = 4.78 𝑠𝑒𝑐Performing the derivation on the equation (1) w.r.t ‘t’ to obtain velocity function:𝑣(𝑡) = 𝑣0 + 𝑎𝑡Plugging in the values:𝑣(𝑡) = 40 + 32(4.77)𝒗(𝒕) = 𝟏𝟗𝟐. 𝟔𝟖 𝒇𝒕/𝒔Hence this will be the final velocity of the ball at impact.Section 1.4:Answer 2:Given:𝑑𝑦+ 2𝑥𝑦 2 = 0𝑑𝑥In order to find the general solution, it is important to separate the two variables x and y andthen perform integration:𝑑𝑦= −2𝑥𝑦 2𝑑𝑥𝑑𝑦 = −2𝑥𝑦 2 𝑑𝑥1𝑑𝑦 = −2𝑥 𝑑𝑥𝑦2∫1𝑑𝑦 = −2 ∫ 𝑥 𝑑𝑥𝑦2∫ 𝑦 −2 𝑑𝑦 = −2 ∫ 𝑥 𝑑𝑥𝑦 −1 −2𝑥 2=+𝐶−12−1= −𝑥 2 + 𝐶𝑦1= 𝑥2 − 𝐶𝑦𝒚=𝒙𝟐𝟏−𝑪Answer 11:Given:𝑦 ′ = 𝑥𝑦 3𝑜𝑟,𝑑𝑦= 𝑥𝑦 3𝑑𝑥In order to find the general solution, we will separate the two variables x and y and performtheir integration separately:𝑑𝑦 = 𝑥𝑦 3 𝑑𝑥1𝑑𝑦 = …

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

Academicwritingcompany.com 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

### Zero-plagiarism tolerance guarantee

The paper that you order at academicwritingcompany.com 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.

### Free-revision guarantee

The Academicwritingcompany.com 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.

### Privacy and Security policy

With Academicwritingcompan.com, 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.

### 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.

## 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:
\$26
The price is based on these factors:
Customer Academic level
Number of pages required
Urgency of paper