Plz help with this First Order Differential Equation.
So the equation is: x*y'-y=x^3-1
I can't separate the variables?
Anyone knows how to solve that?
$\displaystyle x\frac{dy}{dx} - y = x^3 - 1$
$\displaystyle \frac{dy}{dx} - x^{-1}y = x^2 - x^{-1}$.
Now we need an integrating factor:
$\displaystyle e^{\int{-x^{-1}\,dx}} = e^{-\ln{x}} = e^{\ln{x^{-1}}} = x^{-1}$.
Multiply the DE through by the integrating factor:
$\displaystyle x^{-1}\frac{dy}{dx} - x^{-2}y = x - x^{-2}$.
Now the LHS is a product rule expansion of $\displaystyle \frac{d}{dx}(x^{-1}y)$.
So $\displaystyle \frac{d}{dx}(x^{-1}y) = x - x^{-2}$
$\displaystyle x^{-1}y = \int{x - x^{-2}\,dx}$
$\displaystyle x^{-1}y = \frac{1}{2}x^2 + x^{-1} + C$
$\displaystyle y = \frac{1}{2}x^3 + 1 + Cx$.