Solve: xdy/dx=1-y^2

Solution to a differential equation
${\displaystyle x=-ytextandy=x}$

Suppose we have
${\displaystyle \frac{dy}{dx}+f\left(x\right)y=r\left(x\right)}$
and it has two solutions ${\displaystyle y_1\left(x\right)}$ and ${\displaystyle y_2\left(x\right)}$ then how to prove that solution of differential equation
${\displaystyle \frac{dy}{dx}+f\left(x\right)y=2r\left(x\right)}$
Will be ${\displaystyle y_1\left(x\right)+y_2\left(x\right)}$? I think given differential equations is linear first order equation so its solution will be
${\displaystyle y.e^{\int f\left(x\right)dx}=\int r.e^{\int f\left(x\right)dx}dx}$
now do I establish two solution as ${\displaystyle y_1}$ and ${\displaystyle y_2}$ out of this equation?

I have a question on the particular case of the Cauchy theorem for a first order differential equation with separable variables.
$y^{\prime }(x)=a(x)b(y(x))$
If I impose:
$a(x)$ continuous in a interval $I$
$b(y(x))$ continuous and with continuous derivative in a interval $J$
Can I say that there is only one solution in all the interval I that satisfies the condition of passage through a point $(x_0,y_0)\in I\times J$?
Or, instead of the continuity of the derivative of $b(y(x))$, should I impose that the derivative of $b(y(x))$ is limited on $J$?
Can anyone suggest me the correct conditions to impose in this particular case?
Thanks in adivce

determine the inverse Laplace transform of the function.
$L^{-1}\{R(s)\}=L^{-1}\left\{\frac{7}{(s+3)(s-3)}\right\}$

Solve the following first-order nonlinear ordinary differential equation:
${\displaystyle y^{\prime }\left(x\right)+\log \left(y\left(x\right)\right)=-x-1}$

Solve the linear equations by considering y as a function of x, that is,
$y=y(x)$
$3x{y}^{\prime }+y=12x$

A carousel has a radius of 17 ft and takes 28 seconds to make one complete revolution. What is the linear speed of the carousel at its outside edge?