text: 1. ## Undetermined coefficient problem
  
  Find a particular solution of $y^{(3)}+y'=2-sinx$
  
  Solution:
  
  The homogeneous solution is $y_h(x)=C_1+C_2 \cos x + C_3 \sin x$
  
  Now, my first guess for the particular solution is: $y_p(x)=A+B \sin x+C \cos x$
  
  I have all terms repeated here, so multiply by x to all, obtains:
  $y_p(x)=Ax+B x \sin x + C x \cos x$
  
  Would that be the right guess? Somehow the answer that I obtain from this doesn't match with the one in the book.
  
  Thanks.
  
  You're doing great!
  
  Find a particular solution of: . $y'''+y'\:=\:2-\sin x$
  
  Solution:
  
  The homogeneous solution is: . $y_h(x)\:=\:C_1+C_2 \cos x + C_3 \sin x$ . . . . Right!
  
  Now, my first guess for the particular solution is: . $y_p(x)\:=\:A+B \sin x+C \cos x$
  
  I have all terms repeated here, so multiply by x to all, obtains:
  . . $y_p(x)\:=\:Ax + Bx\sin x + Cx\cos x$
  
  Would that be the right guess? . . . . Yes!
  You're probably making simple errors here and there.
  It took me a few tries to get it right . . .
  
  I finally got: . $A = 2,\;B = \tfrac{1}{2},\;C = 0 \quad\Rightarrow\quad\boxed{ y_p \:=\:2x + \tfrac{1}{2}x\sin x}$
explanation: The document contains a math problem and its solution. The user presents a differential equation and asks if their guess for the particular solution is correct. The response confirms that the guess is correct and provides the final particular 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: 1. ## Undetermined coefficient problem
  
  Find a particular solution of $y^{(3)}+y'=2-sinx$
  
  Solution:
  
  The homogeneous solution is $y_h(x)=C_1+C_2 \cos x + C_3 \sin x$
  
  Now, my first guess for the particular solution is: $y_p(x)=A+B \sin x+C \cos x$
  
  I have all terms repeated here, so multiply by x to all, obtains:
  $y_p(x)=Ax+B x \sin x + C x \cos x$
  
  Would that be the right guess? Somehow the answer that I obtain from this doesn't match with the one in the book.
  
  Thanks.
  
  You're doing great!
  
  Find a particular solution of: . $y'''+y'\:=\:2-\sin x$
  
  Solution:
  
  The homogeneous solution is: . $y_h(x)\:=\:C_1+C_2 \cos x + C_3 \sin x$ . . . . Right!
  
  Now, my first guess for the particular solution is: . $y_p(x)\:=\:A+B \sin x+C \cos x$
  
  I have all terms repeated here, so multiply by x to all, obtains:
  . . $y_p(x)\:=\:Ax + Bx\sin x + Cx\cos x$
  
  Would that be the right guess? . . . . Yes!
  You're probably making simple errors here and there.
  It took me a few tries to get it right . . .
  
  I finally got: . $A = 2,\;B = \tfrac{1}{2},\;C = 0 \quad\Rightarrow\quad\boxed{ y_p \:=\:2x + \tfrac{1}{2}x\sin x}$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem and its solution. The user presents a differential equation and asks if their guess for the particular solution is correct. The response confirms that the guess is correct and provides the final particular solution.
  
  Label: question-and-answer