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**chisigma** The DE can be written as...

$\displaystyle x^{2} y^{''} - 2 x y^{'} + 2 y = 1 $ (1)

... so that it is 'Euler's type'. The solution of the 'incomplete' DE is...

$\displaystyle y= c_{1} t^{\alpha_{1}} + c_{2} t^{\alpha_{2}}$ (2)

... where $\displaystyle \alpha_{1}$ and $\displaystyle \alpha_{2}$ are the solutions of the 'characteristic equation'...

$\displaystyle \alpha\ (\alpha-1) -2 \alpha + 2=0$ (3)

... i.e. $\displaystyle \alpha_{1}= -1$ , $\displaystyle \alpha_{2}= -2$. A solution of the 'complete DE' is $\displaystyle y= \frac{x^{2}}{4} + \frac{1}{2}$ so that the general solution of the 'complete DE' is...

$\displaystyle \displaystyle y= \frac{c_{1}} {x} + \frac{c_{2}}{x^{2}} + \frac{x^{2}}{4} + \frac{1}{2}$ (4)

The value of $\displaystyle c_{1}$ and $\displaystyle c_{2}$ can be derived from the 'initial conditions'...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$