text: # Toronto Math Forum
  
  ## MAT244-2014F => MAT244 Math--Tests => TT1 => Topic started by: Victor Ivrii on October 09, 2014, 02:01:12 AM
  
  Title: TT1-problem 4
  Post by: Victor Ivrii on October 09, 2014, 02:01:12 AM
  Find a particular solution of
  \begin{equation*}
  x^2 y''(x) - 6y(x)=10x^{-2} - 6, \qquad x >0 .
  \end{equation*}
  Title: Re: TT1-problem 4
  Post by: Yeming Wen on October 09, 2014, 09:28:21 AM
  Observe that the equation is an Euler equation, then we let
  \begin{equation*} t = \log x
  \end{equation*}
  Then the equation becomes
  \begin{equation*}
  y'' - y' - 6y =10 e^{-2t}- 6
  \end{equation*}
  It is the same ODE from question 3  (http://forum.math.toronto.edu/index.php?topic=445.0).
  Use the particular solution from 3 i.e
  \begin{equation*}
  y=-2te^{-2t} + 1
  \end{equation*}
  Plug in $x$ back, we have
  \begin{equation*} y=\frac{-2\log x}{x^2} + 1. \end{equation*}
  Title: Re: TT1-problem 4
  Post by: Victor Ivrii on October 09, 2014, 10:11:42 AM
  I made some minor editing. Note that \log x, \sin ,\cos รข. to keep them upright and provide a proper spacing.
  
  But what would be a general solution? (I know it was not required in the test). Also I would like to see on the forum solution without substitution: i.e. we are looking at $y_p=y_{p1}+y_{p2}$, $y_{p1}=Ax^{-2}\ln x$ (because $r=-2$ is a characteristic root) and $y_{p2}=B$.
explanation: The document is a discussion thread from a math forum, specifically the Toronto Math Forum, related to a test question (TT1-problem 4) for the course MAT244. The initial post by Victor Ivrii presents a differential equation and asks for a particular solution. The second post by Yeming Wen provides a solution using a substitution method, and the third post by Victor Ivrii provides some feedback and asks for a more general solution without the substitution.
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: # Toronto Math Forum
  
  ## MAT244-2014F => MAT244 Math--Tests => TT1 => Topic started by: Victor Ivrii on October 09, 2014, 02:01:12 AM
  
  Title: TT1-problem 4
  Post by: Victor Ivrii on October 09, 2014, 02:01:12 AM
  Find a particular solution of
  \begin{equation*}
  x^2 y''(x) - 6y(x)=10x^{-2} - 6, \qquad x >0 .
  \end{equation*}
  Title: Re: TT1-problem 4
  Post by: Yeming Wen on October 09, 2014, 09:28:21 AM
  Observe that the equation is an Euler equation, then we let
  \begin{equation*} t = \log x
  \end{equation*}
  Then the equation becomes
  \begin{equation*}
  y'' - y' - 6y =10 e^{-2t}- 6
  \end{equation*}
  It is the same ODE from question 3  (http://forum.math.toronto.edu/index.php?topic=445.0).
  Use the particular solution from 3 i.e
  \begin{equation*}
  y=-2te^{-2t} + 1
  \end{equation*}
  Plug in $x$ back, we have
  \begin{equation*} y=\frac{-2\log x}{x^2} + 1. \end{equation*}
  Title: Re: TT1-problem 4
  Post by: Victor Ivrii on October 09, 2014, 10:11:42 AM
  I made some minor editing. Note that \log x, \sin ,\cos รข. to keep them upright and provide a proper spacing.
  
  But what would be a general solution? (I know it was not required in the test). Also I would like to see on the forum solution without substitution: i.e. we are looking at $y_p=y_{p1}+y_{p2}$, $y_{p1}=Ax^{-2}\ln x$ (because $r=-2$ is a characteristic root) and $y_{p2}=B$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion thread from a math forum, specifically the Toronto Math Forum, related to a test question (TT1-problem 4) for the course MAT244. The initial post by Victor Ivrii presents a differential equation and asks for a particular solution. The second post by Yeming Wen provides a solution using a substitution method, and the third post by Victor Ivrii provides some feedback and asks for a more general solution without the substitution.
  
  Label: question-and-answer