# 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.
The text passage contains a mathematical question about solving an exact ordinary differential equation (ODE) and provides a solution to the ODE.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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.
user
The text passage contains a mathematical question about solving an exact ordinary differential equation (ODE) and provides a solution to the ODE.

TYPE: SINGLE_QUESTION_WITH_ANSWER