Ellen Chang

## Answered question

2022-07-13

What is the standard method for finding solutions of differential equations such as this one? (if there is any)
$x{y}^{\prime }={y}^{2}-\left(2x+1\right)y+{x}^{2}+2x$
where $y=ax+b$ is a particular solution.
Do I substitute $y$ with $ax+b+u\left(x\right)$ and then search for a solution or am I not noticing something and there's quicker way?

### Answer & Explanation

bap1287dg

Beginner2022-07-14Added 13 answers

I think that your hint is "there exists $a,b$ such thaht $y\left(x\right)=ax+b$ is solution". We find easily that $y\left(x\right)=x$ and $y\left(x\right)=x+1$ are solutions.
Hint: Note that your equation is
$x{y}^{\mathrm{\prime }}\left(x\right)-x=\left(y\left(x\right)-x\right)\left(y\left(x\right)-x-1\right)$
Now put $y\left(x\right)=x+z\left(x\right)$.

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