## idku one year ago DE

1. idku

$$\large \displaystyle -y'+y=y^2$$ $$\large \displaystyle -y^{-2}y'+y^{-1}=1$$ $$\large \displaystyle v'+v=1$$ $$\large \displaystyle e^xv'+e^xv=e^x$$ $$\large \displaystyle ve^x=e^x+C$$ $$\large \displaystyle v=1~+C/e^x$$

2. idku

$$\large \displaystyle y=\frac{1}{1+\frac{C}{e^x}}$$ $$\large \displaystyle y=\frac{1}{\frac{C+e^x}{e^x}}$$ $$\large \displaystyle y=\frac{e^x}{C+e^x}$$

3. idku

$$\large \displaystyle -\frac{1}{2}y'+y=y^3$$ $$\large \displaystyle -\frac{1}{2}y'y^{-3}+y^{-2}=1$$ $$\large \displaystyle v'+v=1$$ $$\large \displaystyle v=(e^x+c)/e^x$$ $$\large \displaystyle y=\pm\sqrt{e^x/(e^x+c)}$$

4. idku

So, I can conclude, that: $$\large \displaystyle -\frac{1}{n}y'+y=y^{n+1}$$ has a solution of: $$\large \displaystyle y^{n}=e^x/(e^x+C)$$

5. idku

for all integer n (besides 0) (perhaps not only integer)