What is the solution of the non-linear differential equation?

Question

Suppose that we $y(\cdot)$ is a decreasing and continuously differentiable function with respect to $x$ s.t. it holds

$$(\alpha-x)y^{'}(x)-\beta y(x)y^{'}(x)-y(x)=0$$ that is a non-linear differential equation, whera $x\neq\alpha$ and $\beta\in\mathbb{R}/\{0\}$. Can we solve this by using some kind of a transformation?

Answer 1

$$(\alpha-x)y^{'}(x)-\beta y(x)y^{'}(x)-y(x)=0$$ i suppose that both $ \alpha$ and $\beta$ are constants.

Note that $(yx)'=xy'+y$: $$\alpha y'(x)-\beta y(x)y'(x)-(xy'+y(x))=0$$ And $(y^2)'=2y'y$: $$\alpha y'(x)-\dfrac 12\beta (y^2)'-(xy)'=0$$ Integrate.

$$\alpha y-\dfrac 12\beta y^2-xy=C$$

If $\alpha$ is a function of $x$ (not constant) then this won't work. It will then depend on the kind of function $\alpha(x)$ is.

Answer 2

Hint:

Rewrite as

$$\alpha y'(x)-\frac\beta2 (y^2(x))'-(xy(x))'=0$$ and integrate.