text: # Find solution to exact ODE: $(4bxy(x)+3x+5)\dot{y}+3x^2+8ax+2by(x)^2+3y(x)=0\hspace{0,5cm}a,b\in\mathbb{R}$
  
  So I have this exact ODE
  
  $$(4bxy(x)+3x+5)\dot{y}+3x^2+8ax+2by(x)^2+3y(x)=0\hspace{0,5cm}a,b\in\mathbb{R}$$
  
  I already managed to show that $$F=2bxy^2+3xy+5y+x^3+4ax^2$$.
  
  Now, how can I find the solution $$y$$ ? Seperating the variables is not really an option and it's also not linear. I'm kind of hitting a wall.
  
  The solution to the ODE is $$F=c\implies2bxy^2+(3x+5)y+(x^3+4ax^2)=c$$, where $$c$$ is a constant. This is a quadratic equation in $$y$$ so you can use the quadratic formula to express $$y$$ as an explicit function of $$x$$, if it is really required.
explanation: The document presents a math problem involving an exact ordinary differential equation (ODE). It includes the problem statement, the user's progress in finding a potential function \( F \), and a question about how to find the solution \( y \). The document also provides the solution to the ODE, which is a quadratic equation in \( y \), and suggests using the quadratic formula to express \( y \) explicitly.
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: # Find solution to exact ODE: $(4bxy(x)+3x+5)\dot{y}+3x^2+8ax+2by(x)^2+3y(x)=0\hspace{0,5cm}a,b\in\mathbb{R}$
  
  So I have this exact ODE
  
  $$(4bxy(x)+3x+5)\dot{y}+3x^2+8ax+2by(x)^2+3y(x)=0\hspace{0,5cm}a,b\in\mathbb{R}$$
  
  I already managed to show that $$F=2bxy^2+3xy+5y+x^3+4ax^2$$.
  
  Now, how can I find the solution $$y$$ ? Seperating the variables is not really an option and it's also not linear. I'm kind of hitting a wall.
  
  The solution to the ODE is $$F=c\implies2bxy^2+(3x+5)y+(x^3+4ax^2)=c$$, where $$c$$ is a constant. This is a quadratic equation in $$y$$ so you can use the quadratic formula to express $$y$$ as an explicit function of $$x$$, if it is really required.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document presents a math problem involving an exact ordinary differential equation (ODE). It includes the problem statement, the user's progress in finding a potential function \( F \), and a question about how to find the solution \( y \). The document also provides the solution to the ODE, which is a quadratic equation in \( y \), and suggests using the quadratic formula to express \( y \) explicitly.
  
  Label: question-and-answer