<s>[INST] Solving a differential equation using an integrating factor 
I'm trying the solve the following equation: $ \left\{\begin{matrix}
(x^B+y^B)(xdy-ydx)=(1+x)x^9dx \\  y(-1)=A \end{matrix}\right. $
for $A=1$ and $A=0$.
$B\in2\mathbb{N}_0+1$

My solution is the following, but I got stuck:
$(x^B+y^B)(xdy-ydx)=(1+x)*x^9dx$
$x^B*xdy-y*x^Bdx+y^B*xdy-y*y^Bdx-(1+x)*x^9dx=0$
$x^B*xdy+y^B*xdy-y*x^Bdx-y*y^Bdx-(1+x)*x^9dx=0$
$xdy(x^B+y^B)+[-y(x^B+y^B)-(1+x)*x^9]dx=0$
where:
$Q(x,y)=x(x^B+y^B)dy$
$P(x,y)=[-y(x^B+y^B)-(1+x)*x^9]$
Then I made partial derivation:
$\frac{∂Q}{∂x}:(x^B+y^B)x(Bx^{B-1})$
$\frac{∂P}{∂y}: -(x^B+y^B)-y(By^{B-1})$
And then subtract:
$\frac{∂P}{∂y}-\frac{∂Q}{∂x}=-(x^B+y^B)-y(By^{B-1})-(x^B+y^B)-x(Bx^{B-1})$
But probably I did something wrong and
I'm stuck and not sure where made I mistake.. Can you help me please?
 [/INST] $$(x^B+y^B)(xdy-ydx)=(1+x)x^9dx $$
Duivide by $x^2$:
$$(x^B+y^B)d\dfrac yx=(1+x)x^7dx $$
Divode by $x^B$:
$$\left(1+\left(\dfrac y x\right)^B \right)d\dfrac yx=\dfrac {(1+x)x^7}{x^B}dx $$
Integrate.
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