text: # Verify a solution
  
  • Feb 9th 2010, 10:40 PM
  ChingKing08
  Verify a solution
  Verify that $\displaystyle xy=lny +c$ is a solution of the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$ for every value of the constant c.
  
  I can't figure this one out...I don't even know where to start. Some help would be much appreciated. Thanks!
  • Feb 9th 2010, 10:49 PM
  Black
  Use implicit differentiation on $\displaystyle xy=\text{ln}y+c$.
  • Feb 9th 2010, 10:51 PM
  dedust
  Quote:
  
  Originally Posted by ChingKing08
  Verify that $\displaystyle xy=lny +c$ is a solution of the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$ for every value of the constant c.
  
  I can't figure this one out...I don't even know where to start. Some help would be much appreciated. Thanks!
  
  we need to show that $\displaystyle xy=lny +c$ satisfy DE $\displaystyle dy/dx=(y^2)/(1-xy)$
  
  start from $\displaystyle xy=lny +c$, differentiate it
  
  $\displaystyle y + xy' = \frac{y'}{y}$
  
  $\displaystyle y(y + xy') = y'$
  
  $\displaystyle y^2 + xyy' = y'$
  
  $\displaystyle y^2 = y'(1 - xy)$
  
  $\displaystyle y' = \frac{y^2}{1-xy}$
  
  so the equation $\displaystyle xy=\ln y +c$ satisfy the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$
explanation: The text extract is from a forum post where a user named ChingKing08 asks for help verifying that a given equation is a solution to a differential equation. Another user, dedust, provides a detailed step-by-step solution to the problem. The extract contains both a question and an answer.
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: # Verify a solution
  
  • Feb 9th 2010, 10:40 PM
  ChingKing08
  Verify a solution
  Verify that $\displaystyle xy=lny +c$ is a solution of the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$ for every value of the constant c.
  
  I can't figure this one out...I don't even know where to start. Some help would be much appreciated. Thanks!
  • Feb 9th 2010, 10:49 PM
  Black
  Use implicit differentiation on $\displaystyle xy=\text{ln}y+c$.
  • Feb 9th 2010, 10:51 PM
  dedust
  Quote:
  
  Originally Posted by ChingKing08
  Verify that $\displaystyle xy=lny +c$ is a solution of the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$ for every value of the constant c.
  
  I can't figure this one out...I don't even know where to start. Some help would be much appreciated. Thanks!
  
  we need to show that $\displaystyle xy=lny +c$ satisfy DE $\displaystyle dy/dx=(y^2)/(1-xy)$
  
  start from $\displaystyle xy=lny +c$, differentiate it
  
  $\displaystyle y + xy' = \frac{y'}{y}$
  
  $\displaystyle y(y + xy') = y'$
  
  $\displaystyle y^2 + xyy' = y'$
  
  $\displaystyle y^2 = y'(1 - xy)$
  
  $\displaystyle y' = \frac{y^2}{1-xy}$
  
  so the equation $\displaystyle xy=\ln y +c$ satisfy the differential equation $\displaystyle dy/dx=(y^2)/(1-xy)$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is from a forum post where a user named ChingKing08 asks for help verifying that a given equation is a solution to a differential equation. Another user, dedust, provides a detailed step-by-step solution to the problem. The extract contains both a question and an answer.
  
  Label: question-and-answer