text: # Separable First Order Differential Equation
  
  1. Feb 22, 2012
  
  ### tinopham
  
  1. The problem statement, all variables and given/known data
  
  $\frac{dy}{dx} = y \sqrt{x}$, f(9) = 5
  
  3. The attempt at a solution
  
  $\int dy/y = \int \sqrt{x} dx$
  
  $ln |y| = \frac{2}{3} x^\frac{3}{2} + c$
  
  $y = e^{\frac{2}{3}x^\frac{3}{2}} + C$
  
  $y = Ce^{\frac{2}{3}x^\frac{3}{2}}$
  
  $5 = Ce^{\frac{2}{3}9^\frac{3}{2}}$
  
  $5 = Ce^{18}$
  
  $C = \frac{5}{e^{18}}$
  
  Thus,$y = \frac{5}{e^{18}} e^{\frac{2}{3}x^\frac{3}{2}}$
  $y = 5e^{-18} e^{\frac{2}{3}x^\frac{3}{2}}$
  $y = 5 e^{\frac{2}{3}x^\frac{3}{2}-18}$
  
  Last edited: Feb 22, 2012
  2. Feb 22, 2012
  
  ### kushan
  
  have ln y + ln c
  
  3. Feb 22, 2012
  
  ### SammyS
  
  Staff Emeritus
  Hello tinopham. Welcome to PF !
  
  4. Feb 22, 2012
  
  ### tinopham
  
  Hi SamS, I was going to ask a question, but I was able to solve it. Thanks!
  
  5. Feb 22, 2012
  
  ### HallsofIvy
  
  1) The integral of 1/y is ln|y|+ c, not ln y.
  2) Since tinopham had a "c" on the right side ogf the equation, it is not necessary to have a constant on the left. The two constants of integration can be combined on oneside.
explanation: The document is a discussion thread from a forum, where a user named tinopham posts a solution to a separable first-order differential equation problem. The problem statement is given, and the user provides a detailed solution. Another user, kushan, provides a minor suggestion, and tinopham mentions that they were initially going to ask a question but solved it themselves. There is no explicit question in the main post, but there is an implicit question in the context of seeking verification or feedback on the solution.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Separable First Order Differential Equation
  
  1. Feb 22, 2012
  
  ### tinopham
  
  1. The problem statement, all variables and given/known data
  
  $\frac{dy}{dx} = y \sqrt{x}$, f(9) = 5
  
  3. The attempt at a solution
  
  $\int dy/y = \int \sqrt{x} dx$
  
  $ln |y| = \frac{2}{3} x^\frac{3}{2} + c$
  
  $y = e^{\frac{2}{3}x^\frac{3}{2}} + C$
  
  $y = Ce^{\frac{2}{3}x^\frac{3}{2}}$
  
  $5 = Ce^{\frac{2}{3}9^\frac{3}{2}}$
  
  $5 = Ce^{18}$
  
  $C = \frac{5}{e^{18}}$
  
  Thus,$y = \frac{5}{e^{18}} e^{\frac{2}{3}x^\frac{3}{2}}$
  $y = 5e^{-18} e^{\frac{2}{3}x^\frac{3}{2}}$
  $y = 5 e^{\frac{2}{3}x^\frac{3}{2}-18}$
  
  Last edited: Feb 22, 2012
  2. Feb 22, 2012
  
  ### kushan
  
  have ln y + ln c
  
  3. Feb 22, 2012
  
  ### SammyS
  
  Staff Emeritus
  Hello tinopham. Welcome to PF !
  
  4. Feb 22, 2012
  
  ### tinopham
  
  Hi SamS, I was going to ask a question, but I was able to solve it. Thanks!
  
  5. Feb 22, 2012
  
  ### HallsofIvy
  
  1) The integral of 1/y is ln|y|+ c, not ln y.
  2) Since tinopham had a "c" on the right side ogf the equation, it is not necessary to have a constant on the left. The two constants of integration can be combined on oneside.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion thread from a forum, where a user named tinopham posts a solution to a separable first-order differential equation problem. The problem statement is given, and the user provides a detailed solution. Another user, kushan, provides a minor suggestion, and tinopham mentions that they were initially going to ask a question but solved it themselves. There is no explicit question in the main post, but there is an implicit question in the context of seeking verification or feedback on the solution.
  
  Label: question-and-answer