If \(\displaystyle y^2 = ax^2+2bx+c\;,\) and \(\displaystyle U_{n} = \int \frac{x^n}{y}dx\;,\) Then prove that \((n+1)\,a\,U_{n}+(2n+1)\,b\,U_{n}+c\,U_{n-1}=x^n\,y\)
and deduce that \(a\,U_{1} = y-b\,U_{0}\) and \(\displaystyle 2a^2U_{2} = y\,(ax-3b)-(ac-3b^2)\,U_{0}\)
Another Reduction Integration
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