# 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:

$$Q(x,y)=x(x^B+y^B)dy$$

$$P(x,y)=[-y(x^B+y^B)-(1+x)*x^9]$$

$$\frac{∂Q}{∂x}:(x^B+y^B)x(Bx^{B-1})$$

$$\frac{∂P}{∂y}: -(x^B+y^B)-y(By^{B-1})$$

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?

$$(x^B+y^B)(xdy-ydx)=(1+x)x^9dx$$

$$xdy-ydx = \frac{(1+x)x^9}{(x^B+y^B)}dx$$

$$x \frac{dy}{dx} - y = \frac{(1+x)x^9}{(x^B+y^B)}$$

$$\displaystyle \frac{dy}{dx} - \frac{y}{x} = \frac{(1+x)x^8}{(x^B+y^B)}$$

Based on the form of LHS, take integrating factor of $$\displaystyle \frac{1}{x}$$.

$$\displaystyle \frac {1}{x} \frac{dy}{dx}- \frac{y}{x^2} = \frac{(1+x)x^7}{(x^B+y^B)}$$

$$\displaystyle \frac{d}{dx} (\frac{y}{x}) = \frac{(1+x)x^{(7-B)}}{1+ (\frac{y}{x})^B}$$

Substitute $$u = \frac{y}{x}$$

$$\displaystyle ({1+ u^B}) du = ((1+x)x^{(7-B)}) dx$$

Can you please take it from here?

• Can you please show me how you write it in this form? $\displaystyle \frac{dy}{dx} - \frac{y}{x} = \frac{(1+x)x^8}{(x^B+y^B)}$ Oct 24, 2020 at 20:12
• @PeterF. I edited. Oct 24, 2020 at 20:19
• Thank you, but I do not understand, how you take integrating factor.. Oct 24, 2020 at 20:22
• See if my answer a while ago helps math.stackexchange.com/questions/3879604/…. Oct 24, 2020 at 20:24
• Thanks it is more clear, but I thought that I shoult do integrating factor by following one of the principle on this site math24.net/using-integrating-factor ,,, I was just trying to do it in this way Oct 24, 2020 at 20:28

$$(x^B+y^B)(xdy-ydx)=(1+x)x^9dx$$ 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.