Re: help solving this problem
\(\displaystyle \frac{dy}{dx}-y=xy^{3}\)
This is a Bernoulli equation.
Let \(\displaystyle w=y^{-2}, \;\ y=w^{\frac{-1}{2}}, \;\ \frac{dy}{dx}=-\frac{1}{2}w^{\frac{-3}{2}}\cdot\frac{dw}{dx}.......\text{chain rule}\)
Make the subs into the original:
\(\displaystyle \frac{-1}{2}w^{\frac{-3}{2}}\cdot\frac{dw}{dx}-w^{\frac{-1}{2}}=xw^{\frac{-3}{2}}\)
\(\displaystyle \frac{-1}{2}\cdot\frac{dw}{dx}-w=x\)
Now, we can use a integrating factor since we transformed it.
The IC is \(\displaystyle e^{2x}\)
\(\displaystyle \frac{d}{dx}[we^{2x}]=-2xe^{2x}\)
Integrate:
\(\displaystyle we^{2x}=\frac{1}{2}e^{2x}-xe^{2x}+C\)
Divide by e^(2x):
\(\displaystyle w=\frac{1}{2}-x+Ce^{-2x}\)
Don't forget to resub w:
\(\displaystyle \frac{1}{y^{2}}=\frac{1}{2}-x+Ce^{-2x}\)
Try solving for y if you want to whittle it down further.
There is a nice stepped through example of a Bernoulli. Now, keep this as a template on future ones. Okey-doke?
