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This is first order linear, so multiply both sides by the integrating factorto get
![\displaystyle \begin{align*} (x - 2)^{-1}\frac{dy}{dx} - (x - 2)^{-2}y &= x - 2 \\ \frac{d}{dx}\left[(x - 2)^{-1}y\right] &= x - 2 \\ (x - 2)^{-1}y &= \int{x - 2\,dx} \\ (x - 2)^{-1}y &= \frac{x^2}{2} - 2x + C \\ y &= (x - 2)\left(\frac{x^2}{2} - 2x + C\right) \end{align*}](http://latex.codecogs.com/png.latex?\displaystyle \begin{align*} (x - 2)^{-1}\frac{dy}{dx} - (x - 2)^{-2}y &= x - 2 \\ \frac{d}{dx}\left[(x - 2)^{-1}y\right] &= x - 2 \\ (x - 2)^{-1}y &= \int{x - 2\,dx} \\ (x - 2)^{-1}y &= \frac{x^2}{2} - 2x + C \\ y &= (x - 2)\left(\frac{x^2}{2} - 2x + C\right) \end{align*})
I have the differential equation
>>(x-2)^2 dY/dX - xy + 2y = (x-2)^4
and i need to solve given y(4) = 10. I believe i have to solve it using an intergrating factor?? I have also simplified to:
>> dy/dx - xy + 2y = (x-2)^2 but believe i cannot divide by (x-2)^4 ??
also would the intergrating factor be 1/x??
Thanks in advance
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