text: MAT244--2018F > MAT244--Lectures & Home Assignments
  
  Integrating Factor problem
  
  (1/1)
  
  Ziwei Wang:
  This question is too complicated to solve.... Any suggestions on this problem?
  Find the general solution of [2xycos(y)-y2cos(x)]dx+[2x2cos(y)-yx2sin(y)-3ysin(x)-5y3]dy=0
  
  sushengq:
  My=2x cos⁡y-2xy sin⁡y-2y cos⁡x
  Nx=4x cos⁡y-2xy sin⁡y-3y cos⁡x
  My≠Nx
  
  According to x=(My-Nx)/M
  μ=e(-∫R1dy)
  (My-Nx)/M=(2x cos⁡y-2xy sin⁡y-2y cos⁡x-(4x cos⁡y-2xy sin⁡y-3y cos⁡x ))/(2x ycos⁡y-y2  cos⁡x )=-1/y
  μ=e(-∫-1/y dy)=y
  Multiply 2 on both side
  (2xy2  cos⁡y-y3  cos⁡x  )dx+(2x2 y cos⁡y-y2 x2  sin⁡y-3y2  sin⁡x-5y4 )dy=0
  Let φx=M
  φ=∫Mdx= x2 y2  cos⁡y-y3  sin⁡x+h(y)
  φy=N
  h' (y)= -5y4
  h(y)=-y5+C
  φ=x2 y2  cos⁡y-y3  sin⁡x-y5=C
  
  wenlinwang:
  $$M_y: 2xcos(y) -2xysin(y)-2ycos(x)\\ Nx:4xcos(y)-2xysin(y)-3ycos(x)\\ M_y \neq N_x\\ R = \frac{M_y -N_x}{M} =\frac {-2xcos(y) + ycos(x)}{2xycos(y) - y^2cos(x)}= \frac {-(2xcos(y)-ycos(x)}{2xycos(y)-y^2cos(x)}=\frac{-1}{y}\\ \mu = e^{-\int{\frac{-1}{y}dy}} = e^{\int{\frac{1}{y}}dy} = e^{\int{y}} = y\\$$
  
  both side times y
  $$(2xy^2cos(y)-y^3cos(x))dx + (2x^2ycos(y) -y^2x^2sin(y)-3y^2sin(x)-5y^4)dy = 0 \phi_{(x,y)} , \phi_x = M, \phi_y = N \\ \phi = \int{M}dx = \int{2xy^2cos(y) - y^3cos(x)}dx\\ = x^2y^2cos(y) - y^3sin(x) + h(y) \phi_y = 2x^2ycos(y) - x^2y^2sin(y) -3y^2sin(x) + h'(y)= N\\ =2x^2ycos(y)-y^2x^2sin(y)-3y^2sin(x)-5y^4\\ h'(y) = -5y^4\\ h(y) = -y^5\\ \phi = x^2y^2cos(y) - y^3sin(x)-y^5 = C$$
explanation: The document contains a math problem and multiple attempts to solve it. The problem is to find the general solution of a given differential equation. There are three separate attempts by different users (Ziwei Wang, sushengq, and wenlinwang) to solve the problem, each providing a detailed 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: MAT244--2018F > MAT244--Lectures & Home Assignments
  
  Integrating Factor problem
  
  (1/1)
  
  Ziwei Wang:
  This question is too complicated to solve.... Any suggestions on this problem?
  Find the general solution of [2xycos(y)-y2cos(x)]dx+[2x2cos(y)-yx2sin(y)-3ysin(x)-5y3]dy=0
  
  sushengq:
  My=2x cos⁡y-2xy sin⁡y-2y cos⁡x
  Nx=4x cos⁡y-2xy sin⁡y-3y cos⁡x
  My≠Nx
  
  According to x=(My-Nx)/M
  μ=e(-∫R1dy)
  (My-Nx)/M=(2x cos⁡y-2xy sin⁡y-2y cos⁡x-(4x cos⁡y-2xy sin⁡y-3y cos⁡x ))/(2x ycos⁡y-y2  cos⁡x )=-1/y
  μ=e(-∫-1/y dy)=y
  Multiply 2 on both side
  (2xy2  cos⁡y-y3  cos⁡x  )dx+(2x2 y cos⁡y-y2 x2  sin⁡y-3y2  sin⁡x-5y4 )dy=0
  Let φx=M
  φ=∫Mdx= x2 y2  cos⁡y-y3  sin⁡x+h(y)
  φy=N
  h' (y)= -5y4
  h(y)=-y5+C
  φ=x2 y2  cos⁡y-y3  sin⁡x-y5=C
  
  wenlinwang:
  $$M_y: 2xcos(y) -2xysin(y)-2ycos(x)\\ Nx:4xcos(y)-2xysin(y)-3ycos(x)\\ M_y \neq N_x\\ R = \frac{M_y -N_x}{M} =\frac {-2xcos(y) + ycos(x)}{2xycos(y) - y^2cos(x)}= \frac {-(2xcos(y)-ycos(x)}{2xycos(y)-y^2cos(x)}=\frac{-1}{y}\\ \mu = e^{-\int{\frac{-1}{y}dy}} = e^{\int{\frac{1}{y}}dy} = e^{\int{y}} = y\\$$
  
  both side times y
  $$(2xy^2cos(y)-y^3cos(x))dx + (2x^2ycos(y) -y^2x^2sin(y)-3y^2sin(x)-5y^4)dy = 0 \phi_{(x,y)} , \phi_x = M, \phi_y = N \\ \phi = \int{M}dx = \int{2xy^2cos(y) - y^3cos(x)}dx\\ = x^2y^2cos(y) - y^3sin(x) + h(y) \phi_y = 2x^2ycos(y) - x^2y^2sin(y) -3y^2sin(x) + h'(y)= N\\ =2x^2ycos(y)-y^2x^2sin(y)-3y^2sin(x)-5y^4\\ h'(y) = -5y^4\\ h(y) = -y^5\\ \phi = x^2y^2cos(y) - y^3sin(x)-y^5 = C$$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem and multiple attempts to solve it. The problem is to find the general solution of a given differential equation. There are three separate attempts by different users (Ziwei Wang, sushengq, and wenlinwang) to solve the problem, each providing a detailed solution.
  
  Label: multiple-question-and-answer