$y = {\left( {x + \sqrt {1 + {x^2}} } \right)^n}$ ${{dy} \over {dx}} = n{\left( {x + \sqrt {1 + {x^2}} } \right)^{n - 1}}$ $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {1 + {1 \over 2}{{\left( {1 + {x^2}} \right)}^{ - 1/2}}.2x} \right);$ ${{dy} \over {dx}} = n{\left( {x + \sqrt {1 + {x^2}} } \right)^{n - 1}}$ $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\left( {\sqrt {1 + {x^2}} + x} \right)} \over {\sqrt {1 + {x^2}} }}$ $= {{n{{\left( {\sqrt {1 + {x^2}} + x} \right)}^n}} \over {\sqrt {1 + {x^2}} }}$ or $\sqrt {1 + {x^2}} {{dy} \over {dx}} = ny$ or $\sqrt {1 + {x^2}} {y_1} = ny$ $\left( {{y_1} = {{dy} \over {dx}}} \right)$ Squaring, $\left( {1 + {x^2}} \right){y_1}^2 = {n^2}{y^2}$ Differentiating, $\left( {1 + {x^2}} \right)2{y_1}{y_2} + {y_1}^2.2x$ $= {n^2}.2y{y_1}$ or $\left( {1 + {x^2}} \right){y_2} + x{y_1} = {n^2}y$