Given the equation:
$$x y{\left(x \right)} - x \frac{d}{d x} y{\left(x \right)} - 2 \frac{d}{d x} y{\left(x \right)} = 0$$
This differential equation has the form:
f1(x)*g1(y)*y' = f2(x)*g2(y),
where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = \frac{x}{x + 2}$$
$$\operatorname{g_{2}}{\left(y \right)} = y{\left(x \right)}$$
We give the equation to the form:
g1(y)/g2(y)*y'= f2(x)/f1(x).
Divide both parts of the equation by g2(y)
$$y{\left(x \right)}$$
we get
$$\frac{\frac{d}{d x} y{\left(x \right)}}{y{\left(x \right)}} = \frac{x}{x + 2}$$
We separated the variables x and y.
Now, multiply the both equation sides by dx,
then the equation will be the
$$\frac{dx \frac{d}{d x} y{\left(x \right)}}{y{\left(x \right)}} = \frac{dx x}{x + 2}$$
or
$$\frac{dy}{y{\left(x \right)}} = \frac{dx x}{x + 2}$$
Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \frac{1}{y}\, dy = \int \frac{x}{x + 2}\, dx$$
Detailed solution of the integral with yDetailed solution of the integral with xTake this integrals
$$\log{\left(y \right)} = Const + x - 2 \log{\left(x + 2 \right)}$$
Detailed solution of the equationWe get the simple equation with the unknown variable y.
(Const - it is a constant)
Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = \frac{C_{1} e^{x}}{x^{2} + 4 x + 4}$$